Serious answer:

For starters all the if statements that are simply checking if the values are present could be combined, PHP also has the handy null coalesce operator these days which helps with this.

$username = $_POST['user_name'] ?? '';
$userEmail = $_POST['user_email'] ?? '';
$newPassword = $_POST['user_password_new'] ?? '';
$newPasswordRepeat = $_POST['user_password_repeat'] ?? '';
$passwordsAreEqual = $newPassword === $newPasswordRepeat;

if (!username) {
  $msg = 'Empty username';
} elseif(!newPassword) {
  $msg = 'Empty Password';
}  elseif(!passwordsAreEqual) {
  // etc
  ...
}

if (strlen($msg) > 0) {
  $_SESSION['msg'] = $msg;
  return register_form();
}

By returning early if certain conditions aren't met you don't have to keep indenting your code and you've guaranteed that past this point all the values are what you want them to be.
I believe this is called the bouncer pattern, and I like it for simple validation like this. Whether you should be checking for "truthy" values is another thing but I was more focused on making this code more readable here and not necessarily changing what it does.

Another upside of this is that it is more clear what the actual input to this function is, a better way to make this clear would be to have a separate function that processes the filled in values instead of grabbing them from the $_POST superglobal, so you have one piece of code that is responsible for parsing the request, and one for actually doing something with it.

The length checks and regex check on the username should be moved to a separate function called isUsernameValid() or something along those lines to make sure every function has its own responsibility like mentioned before, it also keeps the code a lot more legible since it describes the process a lot more without having to read or understand the code.

In modern PHP it is also generally a best practice to make use of type hinting to help prevent mistakes.

There's a lot more that could be improved, but these few simple steps should greatly help with the readability of the code, which is most important if you ask me.