Maybe it would help if we knew whether we were talking about tunes or songs-with-lyrics? In the UK, as "song" has words as well as a tune, while a "tune" is just a melody, but in the US a "song" also means a tune.

Tunes are more likely to have various names, because the name is usually just a tag to remember and refer to it, and any other tag is just as valid. Many tunes, for example, have close variants with different names in Ireland and in England/Scotland (and no doubt further variations across the Atlantic). When a tune was taken from its original location, it might become known, for instance, as
"X'x Favourite" or "X's Jig" where X = the name of the player who introduced it there.

Songs (with words) can't really be said to have different names, but there are lots of song melodies that are used for more than one set of lyrics. Likewise there are songs that are sung to more than one tune - many of the broadsheet ballads and other songs didn't originally have a fixed tune, but were sung to any suitable melody known by the singer.

I'm sure there are many examples of all the above, but in there one category in particular that interests someone ?

Marje