Do crosswinds affect which pilot lands the plane?
Never ask a question with "ever" in it - because the answer will very rarely be no!

For example - if one pilot is training, then while they may have planned to do the landing reality might dictate that they have to hand over the controls to the instructor. I'm sure the same will apply to type ratings or differences training - nobody is going to let you have a crack at a severe crosswind landing during your first front-seat flight in an A380.

Additionally, flying schools will often have strict crosswind limits and some may not grant you full "As Pilot Operating Handbook" privileges until you have [x] number of hours AFTER your test.

In other words - you may have two qualified pilots, but only one of which is allowed to land the plane!

But I'm being deliberately pedantic - I suspect you're focussing more on commercial, multi-crew operations rather than two pilots who happen to be in one aircraft and I'm not qualified to answer that bit!

I'm not aware of any requirement to take seat position into account, however.