I think you just don’t know about gay relationship cliches.
The cliche is for gay men, when unencumbered by women will be immediately having wild rampant sex
The cliche for gay women is when unencumbered by men to get serious, quick - like let’s move in now & get a joint bank account! Also the cliche of “lesbian bed death” which is a joke about lesbians in sexless relationships.
So what may seem like a new fledgling relationship to you may not feel this way for your mum.
Also let’s be real that women buying presents is a cliche norm in relationships - as in the men don’t buy gifts, it’s left up to women to do. Women buying other women baby gifts is part of the “sisterhood” - some women just love buying gifts & an excuse to buy baby things! I imagine this lesbian didn’t have kids herself? It’s just fun to gift people things, am sure she meant it to be thoughtful.
Now, you’re living far away & lockdown, so I can imagine even more importance being placed on sending care packages.
I hazard a guess that if you didn’t have any close or keen friends / family, you’d be sat here making a thread all upset about how nobody cares about your pregnancy & wishing someone would make a fuss over you. What is the alternative? Nobody sends you any gifts? You wouldn’t like that, I’m sure...Just accept the gifts with good grace “oh that’s thoughtful, you didn’t need to send that” is all that’s required.
Now as for not allowing your mother to show a photo of her new grand baby to anybody - this is being precious to the extreme. Most people will just smile & nod politely - all babies look the same & nobody is going to take more than a passing look.
With your control over the photos I think you really do need to take a step back & think about what your issue is. Back in my day we used to keep photos of family / loved one in your wallet to show people. I’m not sure what harm is caused by your mum excitedly showing photos of her granddaughter to people ?
But there will indeed be harm caused if you make this into a big deal and tell her she is not allowed to even show a photo to anybody. You will sound unhinged.