I've never seen mention anywhere in the Facebook documentation that a Page can login to a website. To get an access token for a Page it has to be granted by a user, so I'm not sure how the Page would get one otherwise. Additionally, Pages don't have email addresses or other information (first name, last name, etc) that would be required to create the user profile. It's stuff we could ask from the user, but diminishes the point of social authentication.
If you can point me to website you saw that let's Pages login, we'll gladly look into it to see if we can understand what they are doing.
Thanks,
Alex