Claiming ownership of a game is much more than just gaining the ability to change cover art, it also grants access to analytics, and even viewership data through the Drops system, which for privacy reasons has to be kept secure.
Would you rather Twitch just let any random user claim ownership of your game without any checks in place? A domain name (not even a website, just simply registering a domain for your company which you can use as an email address) can help provide a lot of legitimacy to a claim of ownership of a game as the domain will have public whois information showing who owns that domain and usually their company office address (or PO box).