The only fool proof way to tell if a belt is leather or synthetic is to read it on a tag or printed on the belt itself. That is unless you can clearly tell the belt is nylon or fabric or metal or something.
The smell test doesn't work. I have some 100% vegan belts that smell like leather. I've seen some 100% leather belts that smell like plastic. The only way to tell is to read it.
Otherwise, it's just a guess.
With old, questionable belts, there is no sure way to tell. Perhaps a melt-test... but I'd test it out first on a belt that you know for sure is leather and one you know for sure isn't. Perhaps the results are too similar to tell apart, just like with smells.