1. I just visited the /administrator URL of your site (that's why we ask you the URL to your site when submitting the ticket 😉) and it asks me for a username and password, i.e. it works fine.
What you have to keep in mind is that this feature, as documented, simply enables HTTP authentication for the administrator folder. The HTTP authentication credentials are cached forever by your browser, automatically, without confirmation. That's what all browsers do. You can go to your browser's password manager, find the stored credentials for your site, and delete them. Only then you will get the login dialog again, once before the credentials are cached again.
2. Let me turn the question on its head. "Is there a way to use authenticator codes which synchronise across devices?". Yes, there are several. Password managers like KeePassXC (free, open source), 1Password (subscription-based), or even Apple's Keychain (soon to be called Passwords) can synchronise this information across all devices. KeePassXC encrypted data can be stored on Dropbox, OneDrive, Google Drive, and other similar storage providers which means that you can get access to it even if you lose your phone. 1Password can be accessed over the web, without even using an application, making ideal for emergencies. There are plenty more solutions like this such as Bitwarden, LastPass, and so on and so forth.
These solutions are far more secure than receiving a code by email. Yes, we do have code by email as an option on our site, but do remember that we have clients who can just about use Joomla or WordPress enough to maintain a simple site and they're here to get a backup solution to keep it all safe. Some of these folks find using an authenticator app complicated, or don't even have a smartphone (not to mention there are people who deliberately don't have a smartphone).
Finally, I have to note that I have not seen any cPanel feature which would allow you to use any other 2FA method. You'd prefer code by email, I would prefer passkeys as they are far more secure in every way imaginable. Unfortunately, it's up to cPanel to support any of that. When it came to Joomla, I wrote my own code, maintained it for six years, then contributed for free to Joomla when they were in need of a solution like that. I'm afraid doing the same for cPanel is not even remotely possible.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!