If I'm not mistaken, I think many of Zeus' demigod kids never really amounted to anything. So whether Kratos knew who his father was or not, he definitely didn't think he was an equal with the gods in GoW1 and its prequels. He is inordinately strong, though, for sure.
Also, one other reason I think Kratos is actually Tyr is the burn marks of the chains still on his arm. I think, instead of Fenrir biting off his son's arm and making him all mad-angry again, it would be more shocking if he willingly sacrifices his own arm to Fenrir to finally be rid of the chains, and that's why he's only wielding one single-handed axe instead of a weapon in each hand.