You're organising the office dinner party and each employee has brought exactly one spouse (who is not an employee). You wish to seat the attendees alternating employees and spouses, perhaps so that the spouses feel maximally included and nobody starts a work-related conversation.

Sure. I’ll quote from Wikipedia and modify it.

“In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of couples, labeled “A_n” and “B_n” accordingly, at a round dining table so that the A and B alternate and no two couples are sat next to each other such that A_n = B_m”.

Sure. Seating Pokemon trainers and Pokemon, where each trainer brings their Pokemon.

Idea stolen from this video on Gale-Shapley, which is also typically presented in a "gendered / heteronormative" way (but doesn't have to be): https://www.youtube.com/watch?v=fudb8DuzQlM.

I mean there are so many possible examples, right? Anything where you have pairs of partners.. bridge, tennis doubles, etc.