BuddyPress.org

Register.php is a BuddyPress file whereas wp-signup.php is a WPMU file. If you use register.php, then wp-signup is automatically not used. In other words, you use one or the other. To choose to use wp-signup.php, you would need to delete register.php from your theme.

Many people have issues customize wp-signup.php. Therefore, I’d recommend using register.php and customize that file instead.