On one hand, linguistically, either the word mother purely means "person who has given birth", or it also has additional connotations (being maternal, feminine or something, I don't really know)
Not really, the word mother simply means "person who mothers".
The reason why stonewall want to erase the word "mother" from anything to do with parenting is exactly the same reason they want to remove the word "woman" from anything to do with female specific healthcare. To SW, these words can only be acceptable when used in the vaguest of ways, or in porn.
There's a gender Ideology myth, I think it's number 20 the relates to this comment
On the other hand, it's denying adoptive parents the experience of being a "mother", so really, fuck that. It's exclusionary and insulting. Adoptive mothers are mothers!
No one is saying adoptive parents aren't real parents (and the debate around this has been open since forever, and last time I checked no one has been sent rape or death threats for holding the opinion that adoptive parents aren't the same as biological parents...)