Yanbu, op. It totally got on my tits too.
I don't think it is just a saying actually. And, fwiw, I don't think it's about the mil's love for her grandchild at all.
I think it's someone who doesn't quite get the new family situation. It's about the mil's need to be a mummy figure again. It certainly doesn't show respect for the real mummy.
My mil certainly didn't get it. She seemed to think her gcs were her opportunity to relive her parenting days. She was furious when I and her daughter relieved her of this fantasy.
Before that, she had no problem overstepping boundaries, trying to take over and point out my parenting inadequacies with great pleasure. It was a horrible time. I said nothing for ages and then when she said I was lucky my ds didn't have hollow cheeks because I was bfing and that she was going to wean him at 4 months, I exploded.
Op, your mil sounds like she's forgotten herself and is definitely overstepping those boundaries.
Did she go through pregnancy? Labour? Sleepless nights? I'm sorry but nobody has the right to call your child "their" baby. I think it's very cheeky actually. And dismissing you as the mother.
I'd nip it in the bud if I were you. It's easier to do it now than two years down the line.