Emperor Julian who tried to revive the tradition Greaco-Roman religion around 362 AD called it Hellenismos, which is what some modern Greek polytheists identify as.

Other than that there was not a need among ancient polytheists to identify as being in separate religions, religion was more or less synonymous with "ethnicity" or "nationality", a Greek was a Greek and the expectation would be they followed their national/city's traditional religion, likewise with Romans, Egyptians etc.

Over and above that some might identify as a member of a particular cult, as Orphics, Pythagoreans and such.

It is not necessarily the case that there was a hard and fast boundary between religions as such, would an ancient Greek visiting Egypt have a problem getting involved in any Egyptian rituals? Did a Roman who worshipped Isis cease to be a member of Roman state religion? Not in the say way a Muslim today would

In general, polytheism is more fluid than the rigid monotheist belief structures, so cross-identifying or adopting foreign gods does not seem to have been out of place.

It was Christians as a new cult that needed to identify it's members as something other than the state religion of whatever country they were in. If you could ask a Roman what their religion was called "religio" which Cicero describes as the scrupulous observance of tradition cultures.

Another Roman term was "Mos Maiorum" which just means ancestral custom - but again for a roman to say "this is my ancestral tradition" is not different from a Greek or Egyptian making the same claim on their traditions - in the same way, Judaism at that time was just the ancestral tradition of the Jewish people.