Can 3rd Party Keyboards Access Data from Other Keyboards?
3rd party keyboards can't access data entered into the stock iOS keyboard, that's just the way Apple developed the API's.  Without understanding the Text API's it will be difficult to explain why this is.  

If you're concerned about keyboard security, however, you should be more concerned with what info 3rd-party keyboards send to their servers.  These keyboards can send keystrokes, words, and phrases to their developers via the internet.  Granted, not all keyboards do this but every popular keyboard I've seen includes in their ToS that they can send this info if they want.

The "full access" warning message you see when installing 3rd-party keyboards is about doing just that: sending your info to their servers.

This reason is why Apple prevents these keyboards from entering sensitive text.  That would include passwords.  Take a look at this article about keyboard security.  It's a few years old but generally accurate.