It should be now regarded as a legitimately researched line of possible inquiry that turned out to be completely bogus based on spurious correlational observations.
An early enthusiast for this observation was Albert Étienne Jean-Baptiste Terrien de Lacouperie who noted in 1894 in his book
"Beginnings of writing in central and eastern Asia, or, Notes on 450 embryo-writings and scripts":
A closer comparison with plates i. to viii. of the Elements of South Indian Palceography soon showed me that I was on the right track. And a further study of the Vaihu characters and their analysis by comparing the small differences (vocalic notation) existing between several of them, convinced me that they are nothing else than a decayed form of the above writing of Southern India returning to the hieroglyphical stage. With this clue, the inscriptions of Easter Island are no more a sealed text. They can easily be read after a little training. Their language is Polynesian, and I can say that the vocabulary of the Samoan dialect has proved very useful to me for the purpose.
It is useless to dwell on the importance of this little paleographical discovery for the history of civilization, and its dissemination eastwards.
Unfortunately, this guru has kept much of his secrets. No further evidence for this claim appears, but it is equally easy to read the slightly esoteric "everything's connected" theme as a hint.
Q: What's the explanation for the similarities seen between the Rapa Nui script and the Indus Valley Script? … Is it really a coincidence?
No, it is not even a coincidence. While some glyphs, as presented above, may seem similar, looking at the originals tells quite a different story. The number of glyphs that have some resemblance from each scripts is indeed quite low altogether.
The two systems are apart in space, time, culture, language, history, you name it. There is not much connection to be found at all, except that both were developed by human beings and compared by Western scholars. Some of those researchers may have been quite motivated to see something. So much so that when this was first reported they distorted the sparse evidence that was available. A tendency that apparently survived outside of academia.
This is a funny story since sites like "ancient wisdom" still inspire great confidence to the notion that there is a very mystical connection (Easter Island - Indus Valley Scripts). As it is discussed frequently (Quora: How can you explain the similarities between the writings found on Easter Island and the ones found in the Indus Valley (two places 20,000 km away)?) some look at the scholarly debate seems to be instructive.
In the first half of the 20th century a heated debate ensued; examples:
Guillaume de Hevesy: "The Easter Island and the Indus Valley Scripts. (Ad a critical study Mr. Métraux's)", Anthropos, Bd. 33, H. 5./6. (Sep. - Dec., 1938), pp. 808-814:
Thus, I think, enough proofs have been furnished of the biassed spirit and moreover of the levity with which Métraux has dealt here with a scientific matter. As to several other statements of his, no less inaccurate then those I have mentioned his stricture on the above, including Cuna-Script, I rely upon the authority of Prof. von Heine-Geldern (whose superb study of the Easter Island script happily appears in this same number of the "Anthros" determine whether there is substantial of nor them. I leave pos"), to any Likewise, it to the readers of these lines to decide whether a prejudiced attitude may be a sufficient excuse for the way in which Métraux has worked - or not worked.
In any case they will agree that one cannot take very seriously the meddling in science of a person, who, like Métraux, employ such slipshod, frivolous and methods for his studies as I have when the prevaricating shown, especially authority particularly refers, Hunter, regard to whom he viz. declares with that he has read it "with and when another work, disgust"; moreover, to his bound to say that this is the most reckless defamation of a scholar I ever came across." Therefore to take any further notice of Métraux's assertions would be beneath my dignity. I have however felt it was right to this much attention to his "critical study" not only because of his association with such a world famous scientific institutions the Yale but further to Prof. von his statements": I feel authority, Heine-Geldern writes concerning University; put on their guard any future scholars whose honour might be flippantly attacked from this quarter.
You might notice the lack of content in the above that is compensated by a certain level of vitriol despite being published in a supposedly reputable journal?
This idea was early on diagnosed as incapable of holding too much water:
H. Heras: "The Easter Island Script And The Script Of Mohenjo Daro", Annals of the Bhandarkar Oriental Research Institute, Vol. 19, No. 2 (1938), pp. 122-126:
Considering therefore these discrepancies between the two vocabularies as well as the doubtful antiquity of the tablets themselves, it seems beyond doubt that the scripts of Mohenjo Darò and of the Easter Island are not related between each other in any way, and that therefore the similarity existing between the signs of these two scripts is merely coincidental. This is also the final conclusion arrived at, on different grounds indeed, by Möns. Henri Lavacheri who recently undertook an expedition to the Easter Island to verify the statements of, and to study the problems exposed, by Möns, de Hévesy.
But it is of course a limited subset that is presented in the original question. A subset selected to specifically foster pattern recognition in a human mind. Looking at it
Richard E. McDorman: "Universal Iconography in Writing Systems
Evidence and Explanation in the Easter Island and Indus Valley Scripts":
It has been noted that graphically similar symbols have been employed to represent semantically cognate ideas in a number of early, unrelated writing systems. One explanation for this phenomenon is that there exist universal iconographic principles that bear upon the minds of those creating these scripts, thereby influencing the graphical form of the glyphs that comprise the newly created writing systems. The principle of universal iconography as applied to writing systems implies that certain pictorial representative forms tend to be associated with semantically similar ideas irrespective of cultural or societal factors. A result of this principle is that many early logographic writing systems possess structurally similar signs, often representing semantically related referents.
Although the fifty or so characters shared by the two writing systems may be the product of some degree of randomness, at least in terms of the particular forms selected from the universal icon set by the creators of each script, that a few dozen signs appear in the scripts of both the Indus Valley and Easter Island is not coincidence, which implies accidental correspondence with no causal factor. Because both the Easter Island and Indus Valley scripts make at least moderate use of the principle of pictography, the variety of forms exhibited by the symbols in their character inventories is heavily constrained. As a result of this and other factors constraining the graphical forms of the characters that compose the two scripts, between forty and fifty similarly formed glyphs appear in both writing systems. When one considers that the Indus script is comprised of some 500 characters and that the rongorongo of Easter Island number into the thousands, and that the formation of their characters follows a discrete number of definable principles, that the two scripts have in common a few dozen characters is not remarkable.
From Heras some counterexamples for the picture in the question:
Note that the picture in the question seems to compare symbols looking alike. Does it compare symbols that represent also meaning alike?
Some more pictures to compare:
Source, Flenley & Bahn (2003), see below
Another take on Easter Island symbols:
A sample of Indus valley glyphs
Asko Parpola: "Deciphering the Indus script", Cambridge University Press: Cambridge, New York, 1994, p 74.
Why was Hevesy from the first quote so angry at Metreaux?
from: Alfred Métraux: "The Proto-Indian Script and the Easter Island Tablets. (A critical study)", Anthropos, Bd. 33, H. 1./2. (Jan. - Apr., 1938), pp. 218-239.
Trying to get to the root of some development is not a hoax. Proposing a stylised version of a symbol is a legitimate approach to the problem. Not being transparent about deliberate changes however is quite unscientific.
How old is that script that supposedly developed in isolation?
The obvious conclusion is that the ‘script’ was a very late phenomenon, directly inspired by the visit of the Spanish under González in 1770, when a written proclamation of annexation was offered to the chiefs and priests for them to ‘rubriquen en forma de sus caracteres’ (sign by a mark in the form of their characters). Was this their first experience of speech embodied in parallel lines? The document survives, and the marks placed on it (ill. 52) are pretty nondescript except for a vulva, and a classic bird motif on the right which is identical to rock art images and similar to characters on the tablets.
Whatever its origin, the Rongorongo phenomenon now survives only as markings on 25 pieces of wood scattered around the world’s museums. Some signs also survive on paper in makeshift ‘books’ from the end of the nineteenth and early twentieth centuries, but these were considered by the islanders to be an ‘inferior form of script’. The 25 wooden objects contain over 14,000 ‘glyphs’, including one incised staff (which probably represents the original Rongorongo artefacts; the driftwood boards are likely to have been elaborated later on the model of the staffs). They were probably originally called kouhau ta, or ‘written staffs’; another name, Kohau motu mo rongorongo, is a recent invention, translated by Sebastian Englert as ‘the lines of inscriptions for recitation’. This is often shortened to kouhau rongorongo (‘wooden board for recitation’),15 which Métraux believed to mean ‘chanter’s staff’ and hence to indicate a link with Mangareva and the Marquesas where staves were used to beat the rhythm of chants.
The term Rongorongo (chants, recitations) did not exist in Rapanui before the 1870s, and was certainly brought from Mangareva by people who returned after abandoning the Catholic mission there: in Mangareva the Rongorongo was a class of high-ranking experts charged with the memory and recitation of sacred marae chants; it is therefore highly likely that the concept came to Rapa Nui with its first settlers, and the same may well be true of the ‘script’.
All the surviving Rapa Nui pieces are over 125 years old: many look quite unused, and besides, some are fragments of wood foreign to the island, and even include a European oar. It is therefore probable that they all postdate European contact. Although a myth has arisen that the Peruvian slave raids of 1862 removed the last islanders who could truly understand the tablets— knowledge of them was confined to the royal family, chiefs, and priests, and every person in authority was carried off to Peru—this is actually not true. Many of the older people seem to have avoided the raid, but most if not all of them later succumbed to the smallpox and virulent pneumonia brought back by one of the few survivors.
[…] If Fischer is correct, then we now know what most of the inscriptions say, even though we cannot read them yet. But there appears to be a marked preoccupation with fertility, which fits with the phenomenon’s late date. Be that as it may, and whether or not the islanders developed their ‘script’ alone or under outside influence, it remains a crowning glory of this unique culture, one of the most highly evolved Neolithic societies in human history.
John Flenley & Paul Bahn: "The Enigmas of Easter Island. Island on the Edge", Oxford University Press: Oxford, New York, 2003, p173–190.