I think MiMo was right -- the Latin names are also used more or less interchangeably, and literate readers know both and know which are the same characters. But in the context of the (original) Greek myths, the Greek names are probably more common.
At least nowadays. That may not have been as true in, say, the 19th century. If the best-known translations of the Odyssey then used 'Ulysses,' it might explain why that was for a time a first name used in English (cf. the American Civil War general, later president, Ulysses S. Grant), which in turn might explain why Joyce used that version.
You're right that it's probably out of character for a Roman wife to use the Greek name, but hey, it's only HBO, not even PBS or the BBC. I don't watch HBO, so I'm just going by your description, but if it's as much of a costumed soap opera as it sounds like, historical exactitude may have been far from uppermost on the writers' minds. (-;
'Hoi polloi,' however, isn't either a mistake or a sign of particular erudition, because it's a perfectly common phrase in (educated) English. We do normally say 'the hoi polloi,' which IIRC is etymologically redundant, but so few English speakers know Greek that most of us (including me) don't care, though purists sometimes file it in lists with things like 'ATM machine.'
If the writers had wanted to be ultra-historically correct, they could have used something like 'the plebs,' which is also an English word but is to my ears considerably less common, at least, outside thesauruses. Not many (American) English speakers study Latin these days either, much less Roman history.