I'm not sure I would say it is necessarily "illegal" (as in an ADA violation, which it would be for an employer) but I agree it is not a good idea. Even Disney doesn't need to take on the DOJ over something so silly as asking about an individual's disability. It most instances there is no need to ask and in most others a short description of the symptoms tells Disney more than enough.
My point was simply there is an unworkable assumption here. Expecting a public accommodation like Disney to be unable to merely ask - as opposed to requiring "proof" (in all but an extreme case) - when a guest seeks a policy modification is something like telling a physician she can't ask about a patient's family medical history.
ETA: I revisited the DOJ regulations (mentioned above) and its explanation for them. Two points are worth making.
First, criteria that screen out individuals with a disability are illegal (unless the criteria are necessary for providing the goods). One excuse that is per se unacceptable is "The wishes, tastes, or preferences of other customers may not be asserted to justify criteria that would exclude or segregate individuals with disabilities." (I'll provide a cite to anyone who PMs me.)
Second, a DOJ regulation (it has said) "prohibits attempts by a public accommodation to unnecessarily identify the existence of a disability; for example, it would be a violation of this section for a retail store to require an individual to state on a credit application whether the applicant has epilepsy, mental illness, or any other disability, or to inquire unnecessarily whether an individual has HIV disease." This statement is in the DOJ's explanation for a regulation it adopted in 1991, not in the regulation itself.
Don't misunderstand. No one should take this as saying Disney can require proof of a disability. It would, at a minimum, be foolish. I tend to agree with the DOJ's conclusion that requiring "proof" or even unnecessarily asking an individual about a disability would not be consistent with the prohibition against unnecessary "segregation" or even "exclusion."
The DOJ's explanation does mean that Disney could ask a guest to identify the existence of a disability where it is necessary.
(P.S. I enjoyed your Peter Pan photo.)