Generally speaking, I think a person has a right to decide their own name. But if a GP's choice of GP name offends the parents, IMO, the GP should be willing to change it. There are loads of alternatives, surely.
But it's NOT just about the name. It's about this...
"My daughter ALREADY CALLS ME MAMA!!! But she will say tel her “no that’s not mama that’s mummy hawa or even worse she will say “that’s hawa I’m mama”
????????????????! So it’s appropriate for the grandmother to call me by my first name and try take the name that my child already calls me? To call her instead lol?"
... and the fact that MIL refers to her GDs as her "daughters," etc. Clearly, she's having trouble accepting the fact that she not the mum this time around and/or that she's getting older, etc. She can't help feeling that way, but she can help how she acts. It seems to me, she's not only "playing mummy" in her own mind but trying to get everyone, even the kids, to go along w/ it.
So yes, I think you and DH have to deal w/ it, Hawa. Perhaps using humor as Luckygirl suggests, will help. At least, it may make DD see MIL's behavior as "funny" or "silly," so DD won't take it seriously and be confused.
But again, I recommend cutting back on the time you spend w/ MIL. Do you and yours really need to see her twice a week? Unless she's childminding, that's a lot, IMO. Less time w/ her is less time for her to try to confuse DD. And less aggravation for you (I can't imagine having to face this twice a week!) Also, perhaps she'll take your and DH's concerns more seriously if she thinks she lost out a little b/c of her behavior (perhaps not, of course). Again, I'm NOT saying cut her off completely, just lower the number of visits and give yourselves more room .... Just my opinion...