I think yes they can ask for proof since even in casinos the proof is being asked if you are above 18 or not so same way in online itself how will they know if someone under 18 is gambling or not. Identity proof should be must for playing online.
I think this is the best proof, which requires age notification to prove the accountability of its users, I myself have no problem with the request of identity card for them because it guarantees the security in transaction. This also applies to some large market trading exchangers where the id is required for any nominal withdrawal.
I disagree to ask proofs in gambling site. Most of us use bitcoin because It does not required for our identification ID unless we will withdraw a big amount (converting into cash).
The exceptions you mention will be very useful for preventing possible losses on transactions arising from both parties, so you don't have to worry about this anymore.