It'd be easy to do so accidentally. Furthermore, it wouldn't be necessary to attach your MP number to your ticket. If UA is interested in finding you, they will. Assuming you filled out the secure flight data properly -- and, if not, how will you get through security? -- UA would have your name, gender, birthday, and general location. The chances that there are going to be more than one John Smith with the exact same birthday living within 40 miles of an airport are very, very small. And, even if there are, there will likely be enough additional information available to disambiguate them. If the tickets were purchased via the UA website, they'll have geolocation data from the IP address, which will narrow it further.

This goes for all activities that people generally suggest "don't put your MP number on your ticket" (e.g., throwaway ticketing, hidden-city ticketing, etc.). The truth is, companies can identify you anyway.