As someone squarely in gen y. I've had the same feelings about me doing stuff that others are associating with gen z.
I think there may be two things at play here. One is that some people are just bad at adapting to social shifts and assume that everyone is the same way as them. The other is that people have gotten loose with usage of generation terms. So for some older people "gen z" = "person younger than me", while for some younger people "boomer" = "person older than me"
And both of those are problems with the speaker, so now I just ignore them and happily keep on doing the "gen z" things.
FWIW I mean just as a thing that gen-z popularized, I don’t think they think they own the idea (well, I hope they don’t, I’m not gen-z and I use them).
Anyway, the oldest gen-z is just about pushing 30 now, so they get to join us lame people with sore backs.
Thanks for your perspective, I must say that I agree with you