I have basses with different personalities, and use strings as a way to add a touch of further diversity. I just rewired a bass today ewhere (after the change) the neck pickup is the same pickup, same location as another bass - which normally would bug me (I try not to have duplication in the herd), and it did cause me to delay implementing that idea, but....turns out that the combo of a brighter fretboard material, bigger frets (more metal at the contact point), slightly differe pickup heights, and brighter strings in one of the basses made two bases with the same pickup configuration (and circuit) have..noticeably different personalities.
Short answer: Although I try to not have too many kinds of strings around, one kind for all my basses? Not gonna happen.