I understand that ballet and opera companies have huge wardrobe departments. But do dancers ever actually own their costumes?

If a dancer pays for a costume, they own it. When a dancer performs as a guest artist often, it is worth the price of a costume (tutus, for example, can be very expensive). This way, you have what you need when you need it, and what fits and looks good on you. Sometimes, dancers can borrow costumes if it works out that the costume might not be needed, but one can't always count on that....Generally, companies do not like to loan costumes.

When my local dance academy was performing Nutcracker this year, both of our guest soloist brought their own costumes. He had a very nice outfit (from Kirov Ballet), but she had the most tattered dress for the parts before the PDD. I know she was borrowing that one from her company. The tutu looked very new and professional, though, in the end/PDD.

It also depends on the dancer. I know Sylvie Guillem only wears her own costumes.

Male dancers also have stocks of standard costumes. Black waistcoat, white shirt, with or without bow for "Les Sylphides", standard black tunic for Black Swan, all-purpose gray gilet, apple-green tunic, because it goes well with rosy-pink....