This is most likely not a real pattern. "The Romans" did not all subscribe to Pythagorean math-magic or numerology.
The pattern that was observed in the question is not a real pattern:
Nundinae is counting eight days, if we count like we do today. But the Romans counted days inclusively and that is the reason why the etymology is not based on oct- for eight but on non- for nine.
Then we get a calendar reform and the Romans adopt our now familiar seven days week.
Roman measurements of length
Roman measurements of length are also not showing much affinity to the number 8:
625 feet to the stadium, eight stadia to the mile, and three miles to the league
Ancient Roman units of length:
Roman unit English name Equal to Metric equivalent
digitus finger 1⁄16 pes 18.5 mm
pollex thumb 1⁄12 pes 24.6 mm
palmus palm 1⁄4 pes 74 mm
palmus major palm length 3⁄4 pes 22 mm
pes (Roman) foot 1 pes 296 mm
palmipes foot & a palm 1 1⁄4 pedes 370 mm
cubitus cubit 1 1⁄2 pedes 444 mm
sestertius step 2 1⁄2 pedes 0.74 m
passus pace 5 pedes 1.48 m
pertica perch 10 pedes 2.96 m
actus (length) 120 pedes 35.5 m 116.496 ft 60 passus or 12 decempeda
stadium stade 625 pedes 185 m 607.14 ft 600 Greek ft or 125 passus
mille passuum(Roman) mile 5000 pedes 1.48 km 4854 ft 0.919 mi 1000 passus or 8 stadia
leuga (Gallic) league 7500 pedes 2.22 km 7281 ft 1.379 mi
Numbers in the Roman military:
The contubernium was the smallest organized unit of soldiers in the Roman Army and was composed of eight legionaries, the equivalent of a modern squad. The men within the contubernium were known as contubernales. […] The contubernium was led by a Decanus, the equivalent of a junior non-commissioned officer. […]
While a unit of eight "contubernales" does not adhere to the organizational system in multiples of 10 men (“decanus”, “centuria”), when two auxiliaries are counted as an implicit part of the unit, a contubernium does match the nomenclature.
Designating a small unit in the military was by no means fixed across all of the history of the Roman time.
Under Hadrian the contubernium was enlarged to be composed of ten men, in Byzantine time this squad unit would count at 16 men.
Looks to me that the assumption is not based on a real pattern.
This flexibility for a small group of soldiers in an army is even observed today:
US: team (fireteam: 4 or fewer members) < squad (8–14 members)
German Army: Trupp (2–8 members) < Gruppe (8–12 members)
Wehrmacht: An infantry Gruppe consisted of ten men.
That the number eight pops up is just a coincidence as small groups have to have any small number and eight was one of the possibilities the Roman army had the option to choose from and did, for a limited time. Just like any army, they try a few things and stick to that what they think works best. And change that if they get any wiser over time.
Ancient Roman attitudes to numbers
“Ten is the very nature of number. All Greeks and all barbarians alike count up to ten, and having reached ten revert again to the unity. And again, Pythagoras maintains, the power of the number 10 lies in the number 4, the tetrad. This is the reason: If one starts at the unit (1) and adds the successive number up to 4, one will make up the number 10 (1 + 2 + 3 + 4 = 10). And if one exceeds the tetrad, one will exceed 10 too…. So that the number by the unit resides in the number 10, but potentially in the number 4.” (Aetius 1.3.8)
Early philosophers found harmony in numbers. The symbolism and beauty behind each number can be further extended to the essence of all following numbers. The mysteriousness behind the theories founded by Pythagoras and his followers is certainly deeply inspiring and symbolic.
Kate Hobgood: Pythagoras and the Mystery of Numbers
While there might be some features attributable to the number eight,
It has been argued that, as the cardinal number 7 is the highest number of item that can universally be cognitively processed as a single set, the etymology of the numeral eight might be the first to be considered composite, either as "twice four" or as "two short of ten", or similar.
Like a real connection to Etruscan world ages, the eight rays star of Ishtar/Venus and so on. But any mathematical connection is very likely more conicidence than anything else. Although there is flimsy scientific speculation that
It has been suggested that the reconstructed Proto-Indo-European word for "nine" might be related to the PIE word for "new". Based on this, some have speculated that proto-Indo-Europeans used an octal number system, though the evidence supporting this is slim.
Pattern recognition as an explanation
The most likely explanation for assigning a special significance to the "mythical/religious" number eight (or "8") appearing in Roman clusters is the clustering illusion, a phenomenon closely related to pareidolia and apophenia. (Under no circumstances is that to be read as an insult or or an accusation of illness. It's just a human psychological phenomenon that we all share to varying degrees).
The Roman numeral system is based on or influenced by mainly natural phenomena, mesopotamian sexagesimal system and the good old base10 decimal system. A hopefully convincing argument might be made in comparing the actual Roman numeral VIII and the Latin way of constructing numbers with 8 (18: duo-de-viginti… that is a pattern for Roman use: absence of oct- but constructed as "X minus 2") against some so-called Properties of the number 8.
Theoretical counter example adapted to today: "The alphabetical list of the English spellings for the integers 0 throught 1,000 begins eight, eight hundred, eight hundred eight, eight hundred eighteen, eight hundred eighty and so on. The last entry, of course, is zero. How many of your readers can name the 100th, or next to last, number on the list?" –– Does this make the number eight any more significant for English speakers? –– From Martin Gardner: "The Magic Numbers of Dr Matrix", Prometheus Books: New York, 1985.
Further insight might be gained by reading: Dudley Underwood: "Mathematical Cranks. The Amazing Mathematical Solution for Everything", Mathematical Association of America, 1992, p29f.