Well, yes, it doesn't have to be that way; a game could do other things. Like in Skies of Arcadia, poison cancels regeneration and that's on top of the simple damage over time.
But I was speaking mechanically.
For Elise, Elise's Qs are poison in their name, in their flavor text, because Elise is a spider, but they are only poisons superficially. Mechanically, Elise's Neurotoxin isn't any more a poison than Maokai's Twisted Advance is.
So, if someones is being called a quote "Poison Mage" and they are in a video game, what that means, what that combination of words says, is that they are a mage-type character that does damage-over-time to such a significant extent it defines them. Yes, in the actual implementation you could have things like a slow-poison or a stun-poison or whatever. But what I am saying is that is what that term, "Poison Mage", would mean in context of video games.
The other problem with a poison mage is Cass always had a spammy Twin Fangs, even if it was bugged forever.
I reckon that's more a symptom of Cassiopeia's Deadly Cadence.