They are either in an established relationship or just starting something when Finn and Rey are reunited with their respective birth families and discover they're both royalty. Poe isn't. Actually you'd be hard pressed to find anyone further away from being royalty than Poe, whose family is has been working class on both sides since basically forever. Finn and Rey don't care, but it turns out dating a commoner comes with some political/social complications and as much as they'd love to ignore them they have duties and rules now and it's not that easy.

Anon would love a happy OT3 ending but she's mostly here for the angst and politics.

+ if it turns out it's perfectly acceptable and even politically convenient for Rey and Finn to marry eachother and Poe is trying to convince them he's OK with being left out but he's doing a terrible job
++ Finn trying to find legal precedent/loopholes for their relationship while Rey is more blunt and decides to change the rules on her planet
+++ Poe attends some sort of royal gala with them and even though he's his usual cool and charming self outwardly, he's actually feeling very awkward and out of place
++++ Finn and Rey's families are supportive but they can't change the rules just like that
+++++ someone trying to bribe/threaten/blackmail Poe into walking away