You may only connect to your own account or those of a small set of users you explicitly whitelist.
This statement is a little ambiguous, how many accounts can I have explicitly whitelisted?
Could I make an app that allows people to sign up to my service?
I don’t believe the API is suitable for publishing apps at this point, so I would take “small set of users” to mean a private beta test.
No. The section you took that text from is headed: “ The Monzo Developer API is not suitable for building public applications.”