Learn English – the name for the group of words that includes “once”, “twice”, and “thrice”

These are called multiplicatives by Fowler (1850). The term multiplicative is also seen to describe the words once, twice, thrice, and it seems reasonable to distinguish the former as multiplicative adjectives, and the latter as multiplicative adverbs.

The reason English has three different words for those is because English has three different words for 1, 2, and 3. It’s like why we have three different words for sixth, eighth, and twelfth: there’s a suffix here used with regular numbers.

The difference is that instead of ‑th for ordinals, it’s ‑ce for adverbials, and you just aren't recognizing that ‑ce adverbs things — or at least, that it adverbed them once upon a time.

That’s because ‑es was a genitive adverbial suffix in Old English. You can see its remains in all kinds of adverbs that are today spelled with an ‑s at the end.

Note that genitive nouns also end in ‑s. The difference is that making genitive nouns (well, and noun phrases with a clitic ‑’s) is still productive. However, making adverbials this way no longer is so, although now and then people coin new ones along existing models by analogy.

Other ‑s examples of adverbs made like once, twice, thrice include such words as afterwards, backwards, besides, betimes, forwards, hereabouts, needs, nowheres, nights, nowadays, sideways, thereabouts, towards, unawares — with plenty more where those came from. As Janus mentions in comments, there is also a broad set of those which gained a parasitic (=inorganic, non-etymological) final ‑t on top of their ‑es/‑s, such as acrost, against, amidst, amongst, betwixt, gainst, whilst.

Because we made adverbs out of things by adding ‑es (later ‑s) all the time, it was the natural way to make an adverbial out of the numbers. The word nonce has the same origin, although that’s used for a noun not an adverb. In Old English, the word for modern once was ænes or enes, genitive forms coming directly from the Old English word for one, which was án.

By the way, the Old English word án that gave birth to modern one and once also gave us the indefinite articles a, an. But when Old English speakers said ænes (again, that was their word for once), it had two different syllables. That situation was not to last, however, and this led to orthographic changes.

Middle English picked up twice and thrice by the same construction, although with differing spelling and pronunciation than we use today. Then around 1500, as ones became monosyllabic, it began to be spelled -ce to indicate the lack of voicing — and, for the cases of twice and thrice, to indicate a change in character of the preceding vowel.

So what’s going on is that you no longer recognize this fossilized morpheme as meaningful. It means exactly the same thing as those other free morphemes you mention in other tongues.

And no: twice isn't going away. Neither are twin, both, halve, or double. Twain might someday be, though, moving into literary use not bar talk. As for your Italian learners, I strongly advise against saying “two times”; it does not sound right to this native ear. Indeed, the OED specifically reports that twice is:

In all senses now the regular substitute for the phrase two times

On the other hand, tuppence and thruppence are hardly worth a farthing today, eh? :-)