Back not so long ago, it was in the Lily Family, the Liliaceæ, which was quite a diverse family. But DNA studies in this group and all groups have shattered and recombined elements of the traditional botanical taxonomic understanding such that, in this case, genus
Smilax is now the star of family Smilacaceæ, which only has two or maybe three genera:
Smilax,
Heterosmilax, and maybe
Nemexica if the latter is not considered as part of
Smilax. Meantime, grapes are in another genus
Vitis in another family Vitaceæ which, classification-wise, is rather distant from the Smilacaceæ and the Liliaceæ.
Sasparilla, alias Sarsaparilla, the drink, is made from a species or two or three of
Smilax!