No, they don't quite mean the same thing. １）I'm continuing to fight for him. Maybe he is fighting for himself also, so maybe both of you are fighting for him. ２）I'm continuing his fight for him. He can no longer fight for himself, so you are continuing HIS fight. That's how I hear it at least.

Slight difference. #2 is a little more specific regarding whose fight it is.

No. When you ‘fight for somebody’ they are the beneficiary of the fight. But if someone ‘has a fight’ they aren’t necessarily. For example: My mom spent the end of her life creating this library for the community. Now that she’s passed, I’m continuing her fight for her. (She won’t benefit, the community will. It was her fight and I’m continuing it - partly to honor her)