While I don't hate the relative versatility of ships in STO most games with a fleet as varied as this force players to make choices deeper than "this one can use dual cannons and this one can use gravity well"
Take for instance Eve, in which many ships specialize in a specific purpose other than DPS and do it better than any other. It does this by giving bonuses to specific abilities like decreasing power requirements, decreasing recharge times, or increasing potency for certain applications.
STO does this in a diet format with the subsystems but really, a properly specced captain can make up for a +15 to weapons and still get high auxilliary power without crippling his other skills. So many ships become interchangeable and their choice doesn't always mean a lot.
To help alleviate this and add just a bit more specialization to ships, I'd suggest adding a skill or ability bonus on top of that.
Bonuses to gravimetric generator, warp core potential, targetting systems, flow capacitors and such depending on what type of ship it is and what role its intended to fill.
The number of these abilities would scale up to the ship's rank- your ship gets one as captain, two as vice admiral, three for a lock box or fleet ship, or something like that.
In the meantime, captains that ignore their special skill won't be gimped in regular gameplay and can follow whatever career they'd like. They'd just be at a minor disadvantage like a man in a rainbow boat (who would quickly learn their lesson stepping into ESTFs or PvP).