The same problem. Would like to have login and register pades in same style as other site.
There is at least two plugins that trys to add such functionality. Both are not very attractive.
There are just one ‘shutdown’ hook, but it’s called after all the html is done, so your CSS link will be after </html>. Not the right way. You could see such how such method is used in ‘WP-Custom Logo’ plugin ( https://www.wiso.cz/2006/09/21/new-wordpress-plugin-wp-custom-logo-10/ ).
There are also ‘Themed Login and Register’ plugin ( https://www.jameskelly.org/wordpress-plugins/custom-login-and-registration-forms-plugin/ ), but it is also wrong approach: it replaces wp-login and wp-admin code completely. So it ‘doubles’ securiy risk and adds not really necessary code duplication.
In my instance of WP i just added custom stylesheet link in ‘wp-login.php’ and ‘wp-register.php’ pages. Declarations in that stylesheet file overrides standart ‘wp-admin.css’. You coul see it in https://OpFor.spb.ru/wp-login.php (it is localised in Russian, but it doesn’t matter here).
I don’t like that way (to patch WP core files). It would be mutch better if there will be action hook in WP code.
Dear developers! Could you add special hook to html-header code of wp-login.php and wh-register.php? It would be the best solution for login|register pages customization problems. The action hook must be in header code, right after all css (wp-admin.css link and the buil-in style addition).