Greco-Roman mythology is full of many examples of shape-changing. If you look only at Zeus/Jupiter alone, there are countless examples of him changing himself or changing others. Did the Romans have a specific word in Latin that meant "one-who-changes-shapes" like we do with "shapeshifter", or did they express this in other ways?
While I've found examples describing the act, I couldn't find anything used to describe a person or god as a shapeshifter.
The word you want is versipellis.
In a story told by one of the guests at Trimalchio's dinner party in Petronius (Satyricon 62), it's used specifically to describe a werewolf, but it generally means 'One who can metamorphose himself into different shapes or forms' (OLD). It's also used of Jupiter in Plautus (Amphitryon 123).
My suggestion is multiformis. While the dictionary entry doesn't seem have anything like a shapeshifter, this word would be well understood in proper context, with clear and familiar components.
I am not familiar with the Romans having a specific term like modern English does, so adapting other vocabulary for that purpose may well be the only route available.
Versipellis is a good suggestion, but another, attested one is pluriformis:
plūrĭformis , e, adj. plures-forma,
I.having many forms, manifold, varied (postclass.): “pluriformi modo,” App. Flor. p. 341, 14; Mart. Cap. 7, § 729.
