When attuned, my homebrew warhammer is causing all my melee weapon attacks to gain a +2 bonus to attack rolls, no matter which weapon I am wielding, despite the fact that the only modifier I have added to the item is a fixed +1 bonus to melee attacks.

First question: in the Modifier Subtype, which option will give a bonus to attack rolls made with that specific weapon, not all melee attacks? Second question: When I apply a fixed value of 1, shouldn't that apply a +1 bonus, not a +2?

You cannot make it add +1 to the attack only, there are no official weapons that do that, so that modifier does not exist.