What I don't understand is what it has to do with BitBox being a decent HW wallet or not.
Me neither. If you buy the hardware wallet without giving the required info during the purchase, as they ask, there is nothing wrong with it, neither with Trezor. At least that's how I understand it.
From what I see in
aopp.group, though, your wallet sends a signed message to the exchange, which doesn't make any sense.
What Information Will My Wallet Send to My Exchange?
The information the wallet sends to the exchange is a signed message. This message can contain whatever details the exchange considers enough to fulfill their compliance obligations, but only data the exchange already knows, since it is the one who crafts the message.