Log in page

Hi there,
Navigating throw threads I clicked on the fallowing thred here and this pointed out a possible issue. There might be a good reason why the log in page is 100% wordpress ad and hasn’t got the site name on it but I can’t think at any reason. That’s why I am asking you if it wouldn’t be a good idea to “personalize” the log in name with the site title or with the domain name so the user will know that he is registering on a specific domain and not on an wordpress.com/wordpress.org … wordpress.any site
-this can create confusion and password duplicity.
Regards