He could earn twice his present salary at the new job.
Twice two is four.
Merriam-Webster says ‘twice’ followed by a noun is an adverb. In this case, is the noun still called a noun or something else?
The word twice is a numeral Quantifier; it's called an adverb in dictionaries because "adverb" is the traditional wastebasket category -- if you don't know what the hell it's doing there, call it an adverb.
Quantifiers can be moved around like adverbs, though there are special rules for that (Quantifier-Float}, and they are Determiners like articles, so they can modify nouns, too.
This particular kind of numeral quantifier (once, twice, thrice, four times, ...
In the sentence
twice quantifies the Noun Phrase
which is the direct object of He will earn.
Executive Summary: Don't pay any attention to "Parts of Speech"; there's a lot more going on.
1) The word twice is always an adverb.
2)salary and two in the sentences above always are nouns. You can't twist it and call it something else.