I'm speaking from ignorance because I just watched my brother play multiple times. He loves the game.
V and the character-who-I-think-is-a-spoiler are total edgelords, while Dante and Nero are the 90s kid sort of edge where everything's a punchline, everything's exaggeratedly "cool" and everything they do simultaneously makes you want to cheer them on and groan with all your being.
Maybe I'm just misclassifying "edge" and applying it where it doesn't belong. That's certainly possible. But V is TOTALLY edge incarnate. He just walked out of a Hot Topic changing room, talks like he got a tonsillectomy that never healed properly and is accompanied by demons taking the shape of a black panther and a black raven.
As someone who, sadly, hasn't been able to play the game BUT did watch the story in it's entirety along with gameplay in it.
V is edgy yes.....but I say it's the good kind of edge. Since he does have a sense of humor about it, and he even have a pretty silly side to him when you look at his taunt and such. Also, that blue bird you're talking about? He is basically Iago from Aladdin, and that help balance V edgy side by alot.
Also, that bird, tiger & the black golem I assume you haven't seen are enemy from the past DMC games, so they aren't just here for the sake of making him more dark or whatever.
I ain't saying that DMC writting is like shakespear or somethin', but the main character in general have more behind them that one might think at first they aren't just one-liner machine if you prefer, so, don't let appareance fool you, V is a edgy boy, but a good edgy boy imo.