I think that ultimately Picard was Troi's boss and that he was the only reason she on the bridge at all (I've always believed that on other Galaxy-class ships, that third chair may be filled by a different officer or there might not even be
a third chair).
I also believe that the Enterprise
's sciences division has many different departments--medical, stellar sciences, social sciences, biology, and so on--and each had its own senior officer in charge of it. Troi likely wore blue as a result of a background in psychology and that as ship's counselor, she also head of a small staff of junior counselors, IMO.