How exactly does long-term tension (over months) build up in the ligaments, tendons, muscles?
Tension is not a quantifiable thing, rather an abstract concept to describe feelings of stress in muscles.  I think I see what you are getting at but I would separate it into two questions.

"Is massage provably beneficial to muscles under physically induced stress?"

Emphatically yes, however there are so many different types of massage, some are more medically viable than others.  Most massage stimulates muscles to encourage a release of tightening, much like exercise does, though I would assume that there's also a psychosomatic that plays a significant role as well due to human contact.

The other question, more the point "Are massage therapist's claims about relief true?"

Some are and some aren't.  Unless it's by a medical professional (prescribed physical therapy for instance) they can claim whatever they like.  "Reiki contributes to your body balance and wellness by attuning your energy"  This can not be proven, but not really disproven either, so they can claim it all they want though it's purely anecdotal (if not outright fabrication on the part of less ethical individuals.)

In summary: Massage has tangible benefits, but if the claims of the therapist sound too good to be true, they probably are.