There seems to be a lot of bad attitude on this thread!
I actually like that we have tea rounds in my team. Yes, everyone takes their tea/coffee/water a different way, but that's the point. You end up knowing one small, personal detail about a colleague - even if it's it just that they like a splash of milk and 20 sugars.
I don't make the most junior people make me tea. And I certainly don't make the most junior women make me tea. We all make each other drinks because it's a nice thing to do, and we all pull our weight regardless of seniority or gender. It's a little gesture that says despite the hierarchical system, no one thinks they're better than anyone else on a basic human level.
If anyone joined my team and started giving attitude about something as trivial as taking it in turns to get a drink for a colleague, I would start to question whether they were a good fit. It's one thing politely opting out of a drinks round - "Oh, I find this one giant bottle I bring to work every day does the trick for me, so I'll sit out the rounds if that's ok?" to declaring "Not happening!"
I think when you're new, it's very important to observe how the current team interact together and tweak your behaviour to fit with them. Your probation is always part capability and part personality. You need to make an effort to get them on side - obviously without being a doormat. But doing a nice, reciprocal thing for your colleagues isn't being a doormat.
In the OP's shoes, I would join in with the tea round "I'm just refilling my water bottle, would anyone else like a drink?" and then next time someone pops to the kitchen with a bunch of mugs call out, "Ooh, I'll have some water if that's OK - thank you so much!" It may be they've excluded her from the tea round given her reluctance to making tea, but it should be easy enough to get back into it.