It depends on what plastic it is as to if there's anything wrong with a 20yr old sippy cup or not, plenty of families with children spanning decades, using and reusing things, as well as playgroups with original cups still going, but that isn't the issue here, is it.
You're not being ungrateful and your only being a dick if you're nasty about it, and you don't sound like you are.
Your mum isn't a bit of a hoarder by the sounds of it, she is a hoarder for whom a sippy cup your sister never got to use holds emotional meaning and therefore would be great for her sisters baby.
If I'm right then no reason why she wouldn't be suffering from a hoarders mentality and perspective, but I'm sorry you don't have the more 'normal' mum becoming Grandma experience you'd understandably wish for. We cant have everything, and you have a new flat and a new baby to focus on here, not get bogged down by a difficult relationship.
Allow her to gift say one 'special' teddy or similar, as grandma to baby to be, as it will give joy, but make her select which one as a 'special' gift.
Then just thank her each time and say 'sorry no room, am going to have to be really strict about not collecting stuff as we already have too much stuff.'
Be a stuck record. Same message, don't deviate, don't discuss, don't go on about the size of cupboards etc. Keep it really simple.
Most of all don't allow stuff into the car or through the door. If she brings it with her make her take it home again - so sorry mum, just no room.
A hoarders dream is for someone to develop a need and want the stuff they know they need to get rid off that they've saved because it's useful, (though not to them) and can't bear to throw away. Don't hold it against her, just accept it's whats going on in her hind brain whether she knows it or not.
It may seem bizarre to you, but tbh given what you've said about both the poor relationship and her moving abroad I'd say no need for drama or to get into value of items or lack of it, or how self aware she is or isn't. Just create a boundary that keeps things civil, allows you to determine your future, and prevents you becoming the decluttering station.
Once she's abroad I'm willing to bet she wont spend the postage on sending you endless clutter.