How do I theme the user register page? In Drupal 7, we can use a template file like page--user--register.tpl.php, or we can register our own template file.

How we can do it in Drupal 8?
This is similar in D8. Copy page.html.twig from core or base theme and put it into the template directory of your theme.

Rename it like this:

page--user--register.html.twig