Yeah, your bridge officers don't share your limits. Go to the bridge officer trainer in the personnel section of earth space dock, and you can customize your bridge officer skills.
Most level 3 skills require special ways to get them rather than this trainer, but 2-3 engineers with phaser turret 1, phaser turret 2, and either support drone or quantum mortar in the commander slot (this you can do right at the trainer for no special effort) can be a force to be reckoned with. As an added bonus, its also a workaround for boffs being so dumb.
Boffs are notoriously bad for not effectively using their powers. Turret spam away teams aren't smarter, but you can just flood the battlefield with stupid until the cumulative IQ hits 100. As a tac officer, your only real contribution to this tactic is security team, but even a bad player is still usually in better command of their captain than the AI is for the rest of the team.
Actually in the sense of what the OP was asking, yes, they do. Tactical Bridge officers can only learn Tactical BOff powers, Engineering Bridge Officers only learn Engineering powers. Science Bridge Officers can only learn Science powers. You cannot train a Tactical Bridge officer in Engineering powers any more than you, as a Tactical Officer can get a kit with Engineering powers on it.
You, as a Captain, can choose Bridge Officers of different professions to accompany you, but they will only have access to the powers for their own profession. And, after all, a kit is simply a pre-selected group of four powers, which is no different than the four powers each Bridge Officer can access.