I am pp who said my mom is the same, patriarchy remnant(which is not really a remnant as there are so many of them). She claims the same that she loves ds and dd the same, so not true. Few years ago, she went nuts on my dd which was the first time this happened with dd as she is in Europe, so prior I didn't see any of that. Since then, my dd has not visited her alone, not spent any time with her alone. If I am there, I shut down my mom right away and she knows now not to say a word about dd ever. She still complains that I praise dd too much, while she praises my ds like nuts but she will never actually realize this herself. My dad passed away last summer, and he was wonderful to his princess.
Sadly my FIL, in the U.S. is the same bigot like my mom when it comes to girls(and boys being perfect) but even more so! So, dh and I are always supportive of dd as grandma and grandpa(no grandma on the other side) are nasty chauvinist bigots. No way is my dd ever alone with grandma or grandpa. She is now 18 and confident young woman, who sadly realized at 15 what nasty people towards her they are/were.
I recommend you do no have your dd spend time with such a nasty person. It is bad for her self esteem and your MIL will not change. My mom and FIL have not changed, and will never change.
DS himself came to realize that grandpa hates women and is not nice to his sister, as is grandma. He would ask questions like "why does grandpa hate women?" So, the whole thing is absolutely toxic towards both gender kids. FIL was telling DS that women are nasty, problem in the whole world, crazy nasty screws... and sometimes kids don't tell you everything that they are told, and you realize that grandparents are making nasty bigots our of your DS! Hence, ds was not alone with them much at all once we caught on their toxic ways.
You think the damage is only done to your dd? You are wrong.