This is one way to explain how at least three out of four "phasers" are shown firing from the very same location at the saucer underside in "Paradise Syndrome". But it's an odd usage of "bank", both because the episode never mentions that word, and because the other episodes where the concept of "phaser bank" actually is mentioned, it is a piece of technology that does its own targeting, apparently independently of other banks.It is likely that the TOS Enterprise is covered with phaser gun emitters that are powered off her four phaser banks.
If we look at the series as a whole there are examples of multiple phaser banks firing simultaneously through the same emitter points and as you write, independently targeting enemies.
"Arena": multiple banks locked onto Gorn ship. We don't explicitly see all banks firing but each time Sulu fires it's a pair of beams from the same emitter locations.
"Errand of Mercy": multiple phaser banks returning fire but we only see 2 emitter points being used (starboard and forward).
"Who Mourns for Adonais": All phaser banks fire thru the same 2 emitter points. Sustained firing.
"For the World is Hollow and I Have Touched the Sky": Phaser banks 1 and 2 simultaneously fired from 2 emitter points.
Interesting, "For The World Is Hollow..." called out "phaser banks one and two" and is then just referred to as "phasers one and two".
KIRK: Prepare phaser banks one and two, Mister Sulu.
SULU: Aye, sir.
...
SULU: Phasers one and two locked in and ready.
With that example, we can infer that in "The Paradise Syndrome" that "phaser one" thru "phaser four" are phaser banks one thru four and they can fire in sequence and simultaneously through the same emitters just like phaser banks one and two did in "For The World Is Hollow...". SULU: Aye, sir.
...
SULU: Phasers one and two locked in and ready.
Also we have "Balance of Terror" where we see "phaser one" power 3 emitters in sequence before switching to "phaser two".
This is why, IMHO, the four phaser banks on the Enterprise can be used to independently act and fire through any of the ship's phaser gun emitters.
...The refit then replaces each of the quartets of lesser emitters with a single bigger one. Which also means that the formerly invisibly small guns now become visible to the audience!
That could very well be.