I can agree with much of what you say about Mom. My DD has a three year old and is not with the bio Dad. Bio Dad has a GF. GF talks to DD about GS, what he is doing, issues and how they are handling them. She makes suggestions and she listens to what works for us. She is good with him. But she never tries to supplant my DD. She will tell you DD is his Mom. In return both DD and I have told her and bio Dad how much we appreciate her being in GS's life.
I cannot see her ever telling anyone (esp media) that "she" wants to be considered his "Mom." Though we do consider her his "Mom" away from Mom.
(LOL we were at a fundraiser a while back and I was playing Bingo. The three of them came in and asked me if I would watch him for a minute. When I said no I got a lot of looks. Cause I said no there are three parents here, there should be no need for me to watch him. They laughed and went away.)
Point being, though she stands in for Mom, she doesn't try to replace Mom. And that is what I am seeing here. A GF (apparently a short time GF at that) seems to be trying to replace Mom. I don't know if that was her idea or the Dad's. But it could cause a lot of pain and confusion for the kids.