An MPL typerating is an MPL type rating, only valid with the host airline, nowhere el
Not really 8ah.
Only during the course (which ends with the line training) is the MPL limited to the host airline. Thereafter, if you change from airline, you have to do a airline conversion course. But this applies to a normal CPL or ATPL as well. If i'm right the Multi crew Pilot License course, as offered by true blue aviation academy, is an integrated ATP course. So like any other integrated course, you're limited to the FTO and in this case, the airline.
One more thing: i'm interested in the MPL because i'm a Line Training Captain myself and i've experienced that MPL cadets perform well above standard during the line-training. So i wonder where this will lead to.