It’s actually a pretty neat way of conceptualizing it. I’m sure many use it as an alter-ego, but I know there are others that feel it is more of a roleplay instead.
A 'mascot' is really not that far off for some. A lot of furries use fursonas as an idealized version of themselves or who they'd like to be. In my probably biased opinion I think it does a lot more for mental health than you might think. Others use them as OCs and play with them like dolls which I guess doesn't really apply here.
I'm guessing, it's rather the other way around here. People saw OOP potentially portraying their persona onto a dog, by giving it sunglasses and a beer, so it's not just a mascot, it's more of a fursona...