To be honest I'm not sure this was true of every navy in the world, and I only know about it because of C.S. Forester and Alexander Kent. As far as the prize money I get in the game is concerned I just assumed that this was the value of the captured prize when it was sold to private hands, rather than the prize money awarded for its capture. Thats, why you don't get it if you decide to keep the prize for your own navy.

As far as battlefield loot is concerned, that goes into the pockets of your soliders not into your treasury. Again Bernard Cornwell makes it clear that soldiers became rich from the spoils gained on the battlefield, and in particular during seiges, but they also tended to spend it very quickly with local merchants which is why armies had huge armies of camp followers. Not really anything that can be modelled in TW though.