I actually think it's fine to question people on their identity, the only person who can really tell you why they identify as what they identify is them. There's some nuance of course, and you don't want to pry or seem like you're trying to convince them they've used the wrong label, but I think it's fine to ask folks.
I'll try to give a good example. The context of this thread is all of us discussing the difference. It's a bit different than asking a singular person why they may have chose one over the other. A very close person to me chooses the pan label over the bi label because of a reason I personally disagree with and view as biphobic. They choose it because they don't want trans and nonbinary people who subscribe to the biphobic definition to believe they're transphobic or enbyphobic. My response to this is not "that's biphobic, you shouldn't do that." Because the only real conclusion from that would be that I'm saying "You should call yourself bisexual" which would be relabeling them. Now, this specific conversation, it vent on, and all of us having the conversation are on the same page about it all, for the most part. We all agree there is massive, massive overlap between the two and the label doesn't really matter too too much, and all of us would be fine saying we're "bi/pan" probably. If our opinions were more unaligned on the topic, it may have gone differently. Especially if this person said that I am transphobic or enbyphobic for saying I am bisexual, but that's not something they did and it's not something they believe. Also, we did all say that the flag we preferred played a bit of a role lmao. The two of us that are bi love the bi color and the the one who is pan loves the pan colors.
I guess for me, going back to the dating app analogy, I'd rather label myself what I am (bisexual) and have people who have biphobic takes that assume I am transphobic/enbyphobic pass on me than relabel myself to accommodate them. Because I don't think we'd necessarily be able to get along anyways. I think if someone is that opposed to the label found out I called myself pan instead of bi on my profile just to get into their DMs, it would also just push them further into biphobia and they'd unmatch with me anyways.
I think this is why this meme is so hilarious, bi versus pan just sort of carries on and on and rarely leads to satisfying conclusions. But I think there's beauty there, I think the queer experience is so unique and I love to learn about people and also have others learn about me. I love discussions.