Just to add to the above, I don't think you're being over-sensitive as such, but something jumped out at me up thread - one person said "mimic" and there's something to be said for that.
My daughter went skiing recently (sorry!) and accidentally ended up on a red slope and got scared. I asked her what happened/did she cry, and she said yes, she had a bit of a cry, but it was hidden behind her mask, so nobody saw, and in the end it was fine because she just copied the people who were better skiers than her. It seemed like such a great analogy for, well, LOTS of things in life!
You say your colleagues have a lot of "soft skills" that they learned growing up, right? Learn and copy them too.
A friend started work somewhere where everyone skied regularly and went as a group, and he had never been and didn't want to make a show of himself, or say "I can't ski.". So he booked a cheap trip on his OWN to learn in private for a slightly scary and intense week, and then next time the "Oh Dan... we were just talking about Chamonix this year... do you ski?" conversation happened at lunch, he was able to say "yes, I do". And he went with the group that winter and it was fine.
I know you're saving for a house, but some of these things qualify as investments in your career progression. Not buying a horse and having kids so that you can send them to Harrow, but look around at the things people do to fit in, from the chocolate they have on their desks to the places they go on their holidays.
If you WANT to fit in, look at where you can blend (without having to spend £100k!).
Also, get 6 sessions a career coach who knows your industry. I promise it will help. Seriously. You colleagues might have has this kind of support organically growing up, but you can 100% pay for it and get it now. You sound very much like you need a mentor who staddles that professional/how you feel inside divide.