BadBurger
Many “Whelps”! Handle It!
It's worth noting that Daemon is married to Lady Royce of RUNESTONE, which is a castle by the sea IN the Vale, not the Eyrie itself. The Vale is currently being run by Lady Jenye Arryn, something the show doesn't talk about much because it clearly undermines their "these men won't accept a woman ruler, hur dur" narrative.
https://awoiaf.westeros.org/index.php/Runestone#google_vignette
As far as the show is concerned, there is no way that a prince Targaryen very near to the throne would not be the lord of the Eyrie if he married into the Arryns. That would be like Bobby B marrying off Joffrey to some lass from House Fossoway of New Barrell.