They don‘t let me board because my boarding pass details don’t exactly match my identification document (ID, passport, green card)

It should be checked that the data that comes on the ticket or boarding pass match those of the identification document that we are going to show, since the airline can deny us boarding if those data do not match, or some data is missing. 

This is because the Air Navigation Act considers the boarding pass a nominative and non-transferable document. Therefore, the company must comply with the safety standards of access to the aircraft.