A picture featuring my fursona (who is a cat-weasel--the picture involves some commonly blacklisted tags so I'm not linking it here X3) has been tagged domestic_dog and canine and I'm not sure if I should go and correct the tags or just leave them be. When I designed him I did joke that he kinda looks vaguely dog-ish, so I get the confusion.

It's sort of both. Obviously, we can't just tag a character as a parrot when they're clearly a horse, but then, considering the tendency of many furry artists to use the same basic body structure for totally different species, sometimes a bit of TWYK is necessary, within reason, of course. In your case, go ahead and change the tags, using domestic_cat, weasel, and hybrid.

It's also not a bad idea to put a note to that effect in the character Wiki entry, and mention reasons when doing tags that might not be immediately obvious why you did. Like, I posted a note to that effect when a guy behind the curtain was not really easy to see. Just the edge of their body was shown as they peeked around it. I don't think I even tagged gender because it wasn't part of the main scene? Not sure - I likely didn't even think of it.