They don't consider the key to belong to the user. The key is a token generated by the site to allow it to identify a user. In order for them to do perfectly so they do not want users to be able to tamper with them, leak them, or do anything which might violate their assumptions about the key.

No, the private key is generated and stored client-side, and never sent to the server. Even if that wasn't the case, how I store my credentials is none of the websites' business, and my own hardware should do as I say.