Should I escalate this?

Should I escalate this?

I have a very condescending coworker who seems to think that he's my manager; he has been with the company the longest in my department and is fairly indispensable, but as far as I'm aware, he's not in charge. He has even complained on occasion that he feels as though he should have "senior" added to his title because of his seniority, further leading me to believe that even he knows that he doesn't have any managerial responsibilities.

My actual manager is very flexible in terms of WFH, but I was recently questioned about my productivity by this coworker. He started asking if I "had enough to do". He also questions me pretty frequently if I leave early for doctor's appointments (I have a medical condition requiring specialist visits and a two-year-old who recently had an ER visit), in spite of the fact that he himself usually leaves to "WFH" after lunch. I've been completely forthcoming with my manager about my personal life in this regard, but things like my medical diagnosis are frankly none of his business.

I have some bias against him because I don't especially like him as a person (mainly because of stuff like this), so I want to make sure I don't escalate this just because of that.

Would it be inappropriate to ask my manager about the chain of command and then discuss with him my concerns about this coworker overstepping? There is also the possibility that I've misunderstood, and he is someone to whom I should report.

Another consideration: I've been with the company for less than a year, so I don't want to make waves so soon after being hired, but I obviously don't want to feel like I'm answering to a colleague.

This workplace is pretty male-dominated: there are literally NO women in managerial positions. I brought that up in a casual conversation I was having with him and another woman, and the woman insisted that none of the women working there were interested in leadership roles. The other woman in our department seems content to defer to him, so the company culture just may not be right for me. I don't want to leap there yet, though, because the flexibility in my work schedule is absolutely indispensable right now.

Anyway, that's a lot. Let me know what you think!