For anyone interested in how it all works. So the login should be legitimate even if it’s on a third party site.
Now it seems you will have to contact Blizzard before putting it on your site, and probably form a contract with Blizzard. What that contract entails, if there’s a specific amount of time it’s allowed for, etc. no one but Blizzard and the site makers would know.
However, again, Blizzard can’t give you any details. You would have to ask the people who own that website.