[@Ambrosia] Aphrodite, or whomever, could use whatever name they wanted on Earth - it is more than likely that, when encountering another of their kind at least, they would use their 'actual' name rather than an alias. So they'd call one another Ares, Balder, whatever, but may well use a completely different name the rest of the time.

Then again, only another deity would be able to 'see' them so-to-speak, for who they truly are. Any mortal who saw it would likely go blind, or not...depending on things.

They probably would be switching identities, although not necessarily have to, especially in places where same-name offspring are common, such as having Artemis be Julian Rogers the Sixth or some such. Yea, they'd look near identical, but I've no doubt a former god would have ways of keeping people from knowing they're one and the same.

I hope that answers it, I do like to ramble.