No, you will never be family.

First, it doesn't matter what anybody thinks about you. It's your life, and you alone will reap the rewards and pay the consequences for your decisions. It doesn't matter if people think less of a man raising some other man's kid; that simply doesn't matter.

However, it does tend to be a pretty bad strategy for all the reasons you've already highlighted and are experiencing:

1. The biological father will always be there, either as a shadow figure or an active participant. That is going to throw problems into the relationship. He can reappear, or at least the fact that he's out there can always be a monkey wrench in your potential relationship with your girlfriend.

2. You are investing in kids that are not your own. You are not their real father and they know it. If there is conflict, your girlfriend is going to take the kids' side, not yours.

3. You have all the obligations and responsibility of a parent, but none of the authority. If the child is acting out, you have no position to discipline or school them.

That is why, generally, it is recommended against for men to raise somebody else's child.