My ex-FIL did this with my DS when he was a baby. My DS' name is the longer version of another name in it's own right, and for a few weeks ex-FIL insisted on calling him by the other, much shorter, version of his name.
I can't think of a male example for this, off the top of my head, so... we'll say that DS' name is "Elizabeth" (it's really, really not!), and his nickname has always been "Beth". Ex-FIL was insisting that his name was "Eliza". And then calling him "Liz". For short.
It was a control issue. Ex-FIL had always been allowed to dictate everything that went on in the family, and no one dared to stand up to him. My ex-partner is still very frightened of him (works for him, lives in a house owned by him, drives cars which he's bought "for" him but takes back at a moment's notice...), but I wasn't. And I stood up to him. Every time he called my son "Liz", or "Eliza", I'd smile serenely and point out to whoever he was talking to that DS' name is "Elizabeth", or "Beth" (it's really, really not!). It was, after all, a name which ex-partner, DD and I chose together (although weirdly enough, there's never been an issue over DD's name... probably because it's the same as ex-SIL's... which ex-FIL apparently chose!). It resulted in ex-FIL sulkily referring to my DS as "Elizabeth" - but that's never been a problem. Using his full and correct birth-name is never going to be an issue as far as I'm concerned - but deliberately calling him something else just because he'd decided he had the right to name his first grandson over the parents...? Not going to happen. DS is 11 now, and ex-FIL attempted to reassert his authority a few months back by calling DS "Liz". DS shot him down, apparently, and saying that he doesn't like that shortened version of his name, he's always been known as "Beth", and he would prefer it very much if his grandfather would call him that, instead, please. Ex-partner backed DS up 100% according to my very bemused ex-MIL...
Names are important. They become our identity, how we become "seen" by other people. They are also the very first gift which our parents bestow upon us. No one has the right, in my opinion, to alter a child's name other than the parents, or the child themselves as they grow older. Yes; your DD will have friends who'll alter her name in all sorts of weird and creative ways - but those nicknames will only stand, if your DD wants/allows them to.
OP, stand your ground over this. Your MIL already named at least one child. Now it's your turn to name yours.