I'm sure you can bring your own water but why go to all that trouble; the tap water on the ship will be cleaner and taste better than any you could bring.
Yes it is safe to drink. I "made" potable water in the Navy. Shipboard water is tested for bacteria, filtered, and treated if needed. This cannot always be said for the bottled stuff.
We drink tap water at meals on the ship, in the cabin, etc. We don't buy the bottled stuff. That leaves more money for beer.
I drink water at meals also, but I do bring bottled water to take with me when we port because I have a medical condition that causes me to dehydrate faster them normal. And I am afraid of being caught somewhere that I do not have access to bottled water.