Does resting fish before serving affect its texture or taste?
Would resting fish change the its taste or texture in any noticeable way?

No, in short.  The only mention of 'resting' of any fish is from the Lophius genus (aka Monkfish, Angler fish).  It is suggested to allow the monkfish to rest if you are going to slice it before serving to reduce the amount of liquid released.  

I would suggest to portion the fish before cooking, and serve asap after cooking.

I think mainly the term 'resting' for fish is inappropriately used, as the fish is actually undercooked and the carry over/residual heat completes the cooking before it is consumed.