In the Arcachon Bay, there is an island called "île aux Oiseaux". I was quite surprised that aux is used as a preposition instead of de or d'. Why is that?
✔ Although "l'Île aux Oiseaux" became a proper name (as the capital Î and O suggest), "aux" originally means "avec des" in this case.
~ The island with birds (on it).
See https://www.cnrtl.fr/definition/au III D. 3. b)
la chambre nue aux persiennes closes
les routes aux vieux calvaires
✔ "L'île des Oiseaux" is the contraction of [ île
de les Oiseaux ]
~ island of the birds = these specific birds, definite
It can be a place name with the same meaning as "aux oiseaux". Similarly, there is an "île des Faisans".
ferme de Rouge-Sel
domaine des Sept-Pendus
✘ "L'île d'oiseaux" is the contraction of [ île
de oiseaux ] and would mean the island made of birds (indefinite) which can be a poetic thing but not the meaning we're looking for.
Au milieu du lac flottait une île d'oiseaux immobiles.
It could not have been
l'ile d'Oiseaux because oiseaux is in the plural, if we were talking about a single bird then indeed it could have been l'ile de l'oiseau.
So your question is Why île aux Oiseaux and not île des Oiseaux?
Using de1 would imply that the island belongs to the birds, obviously that's not the case. The island is a refuge for migratory birds, it's a place where there are lots of birds, hence its name.
About the use of de there's a good answer there.
1 Reminder: des stands for de les.