I'm unable to confirm your issue using a Date field set to Age display on CB 2.0.5. Please update to CB 2.0.5 if you haven't done so yet and confirm if issue persists. If issue persists disable non-core CB plugins 1 by 1 until the issue goes away as a plugin could be altering the edit display of the field.

Another possibility is a fatal jQuery error on your registration page as the date field is purely jQuery (generated off a text field). Press F12 in Chrome or Firefix, click the Console tab, then refresh the page and look for any fatal (red) errors.

Profile Edit and Registration field output code are basically the same. I tested registration as you stated that's where your issue was. I could not confirm. Your issue is likely due to something loading on registration that's causing a fatal JS error. You need view the Console tab of your developer window, which is opened by pressing F12. That's the only way you're going to see if a fatal JS error is present (very likely), which would break the data field jQuery from turning the text input into dropdowns. Your screenshot is of the HTML structure and not the console.

The first error, the syntax error, is what's doing it. It's an unclosed line so any code after it is basically invalid. No idea the source as it says your main.php is the source, which could be from your Joomla template. Change to default Joomla template and see if the issue goes away.