"Multiple" sounds more natural to me.
Both are correct, or at least colloquially acceptable.
Both mean about the same thing, but "many" seems to imply (to me) that you have to hear the name a number of times which is probably equal to more than 4, (but that is just my assumption, others might interpret this vague terminology slightly differently.)
As a native English speaker, (from Utah,) I use "Multiple times" and was wondering if this is correct, because I know the word primarily from math class, but I know other people use it this way. To me, it sounds more natural to say "multiple times," and as someone else said, "many times" would imply a greater amount than "multiple times" would tend to, although both are vague. "A few" is definitely 3 (or approximately 3.) People use these terms very loosely, it is not a big deal.
If "Multiple" means "containing many subparts" then it is still correct to use it this way, since it contains multiple instances of the same thing, that qualifies as subparts of the whole experience which is a combination of similar events which are being pointed out.