So my apartment complex was sold, and I and several other tenants have cats. The original owner allowed them, but the new owner has given us notice to move or get rid of the cat(s). Is this legal?

Check your lease agreement. Most leases will have a clause about pets–whether they are allowed or not allowed. The default is usually "not allowed" unless there is a pet addendum incorporated into the lease.

The lease terms transfer from old owner to new owner, so unless you signed a new lease with the new owner, whatever is in your original lease is valid. If your cat is included in the lease agreement, then the new owner may start a no-pet policy with new tenants but your cat is "grandfathered" in under the original lease terms.

If the cat is not part of the lease and you wish to stay, my recommendation would be to ask your new landlord to do a walk through of your unit and see the condition in which it is kept. If you keep house well and there is no evidence the cat lives there, you may be able to bargain for a concession. However, what one tenant is allowed is usually expected by the other tenants, so it's possible the new owner just wants to have a pet-free building.