As T.E.D. already mentioned, titles were tied to the territory, and mostly didn't change unless a feudal lord higher up in the "foodchain" granted one of its vassals a higher title (it usually came with further land and possessions as well).
Also, once you fulfilled certain requirements to create a title, you could do so (great example, the British Empire, which was created after it annexed other titles on the same level as kingdoms in India).
As for Italy, there were various kingdoms throughout its history.
Before Rome became a republic, Rome was ruled by several kings and after the fall of the Roman Empire the land was split among many factions, among them kingdoms. Byzantium got some of southern Italy later on, but first both Odoacer and Alarich of the Visigoths founded kingdoms in Italy. They were lastly supplanted by the Longobards, who ruled most of Italy either as dukes or kings.
They were then mostly conquered by the Carolingians, namely Charlemagne, which put an end to titles of the level of "king" in Italy for a long time. It also marked the annexation to the frankish empire later HRE. (You mentioned Savoia, which initailly was part of burgundy by the way). The only exception was the papal state, which to this date still is a monarchy of "king-level", where the pope still holds the title of king of the vatican and bishop of the 'holy see' -which are technically two separate titles with two separate functions (forgive the technicality, but that's what we're talking about here anyways - also check out the video about the vatican by cgp grey, it's great at explaining these title relationships).
In the following time, most dukes, counts and marquis in Italy was de iure (by right, i.e. formally) a vassal of the HRE but many were de facto autonomous, sovereign states. Exceptions here are the muslim emirate of sicily (which was later conquered by normans and became a sovereign county), the papal state and the republic of venice, which were all independent states. Sicily however retained a de iure kingdom status, but no one claimed the title. Sardinia also, after some trouble between Pisa, the Pope and the kingdom of Aragon, gained the title of kingdom. And that's important.
Now as to Savoia: it started off as a county, became a duchy (still imperial) as you already mentioned. After the war of Spanish succession it gained Sicily, which had (see above) the formal possibility of being declared kingdom. It was swapped for Sardinia, which too had the formal title of "regno di Sardegna". The Duca di Savoia actually already had some formal titles of kingdoms, remainders of the crusades, but the Duchy of Savoia still was his main title. After gaining the kingdom of Sardinia, he officially became king "Re di Sardegna", though his primary title still was "Duca di Savoia". In the 19th century the title was merged, creating the kingdom of Piedmont-Sardinia. Note that after the restoration the only other peninsular kingdom beside the papal state and the kingdom of Piedmont-sardinia was the kingdom of Two Sicilies, which also traces back it's "titolo regio" to Sicily being a king-level possession.
Long story short: he didn't swap Sicily for a titular difference, but for practicality: it was closer thus easier to rule. Also, technically most italian States were vassals of the empire, but only de iure, because the emperor always had problems controlling italian vassals. Italians were very unruly: anyone who claimed it had so many problems with it that it either lost the territory or just said "meh, i'll let you do as you please, just as long as you formally belong to me". That goes for spaniards, french, austrians and even the HRE.
I'd like to add, that being a duke of the empire, the duke of Savoy still had a voting right and a place in the imperial Diet, though he never made use of it.