Feels Weird Being Called "New" Name For First Time

So today I decided to come out to my social worker. She's really nice, and I've known her for years. Of course, she's fine with it. I didn't expect to see her today to be honest, as I was suppose to see her last week but she had something else going on. So there was no mental preparation for this.

So I just said I was non-binary at first because I didn't want to confuse her or anything. And then she was just asking pronouns and I said they/them, for simplicity sake. I really would prefer Niko/Niko's in a perfect world. And then she mentioned that other people whom she's worked with said they are a specific type of non-binary and I said I was, and explained that androgyne (don't know how to pronounce this so if anyone does, it'd be appreciated. Dictionaries pronounce it different ways) is me feeling male and female at the same time, like it's one gender and not two different things for me (I'm not genderfluid). And I mentioned I chose a name I'd like to be called, and she asked what I was, and I said Niko. And then she said she liked how it came from my original name of Nicole, since I do like my name, and asked if she could call me Niko. I said it was fine, and so she did. I liked it, but it was hard for me. For years, I've been called Nicole, so having this person who has known me for years call me by my name instead of my birth name feels... weird. Is this a common experience? I really like the name, and think about it a lot, and have read this thing where you could test your pronouns and name to see if you like it out loud, and it sounds good. But hearing it out of someone else's mouth felt unfamiliar to my ears. I felt like a different person. I was more confused than not. I've never hated my name, so maybe it's because of this or maybe just that I feel people judge me for this? I have social anxiety disorder, so I feel constantly judged 24/7 for anything. Maybe I need time like others need time.

Tl; Dr.
I came out to my social worker and she asked if she could call me by my name instead of birth name. The experience felt weird. I like my name, but hearing people who I've known for years say it makes me feel anxious and uncomfortable. I've been called by my birth name for so many years that I have attachment to it, as I've never hated it in the first place. I just want a gender neutral name. Has this happened to other people who have changed their name?

 

Thank you everyone for your response.

I feel like things will also get easier once I move away from home. I'm going to university this year, and university seems like a "clean slate" so I can just introduce myself as Niko and not worry about it.

I won't be called by my preferred name for a while though, as I won't see this social worker officially ever again. Today was my last appointment. And the other person I came out to I don't really see in a personal setting.