Desalination is hardly new or cutting edge technology, and I would be amazed if even order Chinese warships don't have them as standard.
I do believe that when a warship connects to an UNREP ship, they can take on water as well as fuel. The reason for that, as well as the UNREP ship itself taking on fresh water at port is twofold.
Firstly, desalination plants are notoriously expensive to operate, that is why they have not really taken off commerically anywhere except for a few super rich Gulf states who have cash to burn. I am sure the warships were all designed to be fully self sufficient in water from its onboard desalination plant, but it run the plant at those levels would be extremely costly. So it just makes far better economic sense to buy fresh water at port and then pump it over to the warships as they refuel.
Secondly, warships are warships and not cruise liners. When I talked about being self sufficient in water from desalination plants, that would not be in terms of everyone being able to use as much water as they like. I would imagine drinking water won't be any issue, but for secondary uses such as washing, cleaning etc, water use would need to be rationed to make sure there is enough water available for essential needs of the ship and crew.
If they get water from port and also top up from the UNREP ship during resupply, those limitations on water use could be greatly relaxed or even waved altogether. This may seem like a small think now, but when you are deployed for months on end out at sea, that would make a big difference to your quality of life and in turn boost moral.
On a side note, water from desalination plants would be perfectly drinkable, in fact, it would almost certainly be cleaner and purer than what you or I might get out of our own taps at home because the desalination process itself requires a much higher level of filtration and purification than what your average mains water purification plant would employ (hence why desalination plants are more expensive to build and operate).
There would be little point in having desalination plants otherwise because a crew can go without washing or use seawater for that if pushed, but they cannot drink sea water. The water they buy at port is actually almost certainly to be the stuff they use for none drinking purposes like washing, cleaning etc because:
1) It would be cheaper than desalination or bottled water.
2) It would be more efficient. They would just buy mains water, which would be far easier and quicker to take onboard and fill their storage tanks with, rather than pallets and pallets of bottled water which would take more time to load and transfer to the warships, and which would also take up additional storage space better used for dry consumables.
3) It would be safer. Not all countries have the same standards when it comes to mains water, and even in many parts of China, mains water is not safe to drink straight out of the taps, and need to be boiled first. If you take drinking water onboard from foreign ports, there is little control you have over the quality and safety of that water, and it could even pose a security risk as an enemy or terrorists could potentially poison or contaminate that water and incapacitate or kill many members of your crew before the sabotage was discovered. Even if the water was uncompromised, you would still want to at least boil it before allowing the crew to drink it. That takes a lot of energy to do, and you also actually loose a great deal of that water in the form of evaporation in the process.
