Am i being a petty bitch?

[My SO and i have been together for 10+ years. We aren’t married but there were talks before about it but it’s basically died out. ( we both essentially wanted to get married) However, people constantly refer to me as his wife and shit and he doesn’t correct them but yet doesn’t seem like he he’s in any hurry to actually get married...However, every instance where I’m referred to as his wife or Mrs *his last name*, i say “we aren’t married” or “I’m not his wife”. And i can tell he gets a little agitated by it...Is that being petty? I just don’t want to walk around claiming a role or allowing people to label me as someone I’m not. It annoys me and he knows this. If he wanted me to be his wife, I’d be his wife right? why tf should i answer to that when it’s not true...I’m 30 years old with no kids and i think I’m just tired of this “significant other” and “girlfriend” thing....i honestly feel like I’d rather just be alone instead of being embarrassed and being labeled. It’s like, what are we even doing? What is all of this for? ugh, it’s so frustrating as I’m trying to put my thoughts into words.