This could well depend upon where you live. It's commonly known to we 'Westies' as the South China Sea, but perhaps the Vietnamese call it the East Vietnam Sea. Then again, the people of Borneo might call it the North Borneo Sea...and, as you mention, may it is rightfully called by the people who reside in the Philippines as the West Philippine Sea. It's certainly not a claim to sovereignty, whatever it is called.
Take a look at the North Sea, which lies between England, Scotland, France, Denmark and Norway, with Iceland off to its west. It's called the North Sea by the British but I'd feel pretty sure that the Old Vikings out of the Norse Lands would have called it something different again.
This is why around a hundred years back - after an international meeting - it was decreed that a country only owned that water surrounding it up to a three mile limit. Then, of course, money got in the way and nations started claiming twelve mile limits even over straits and channels which were less than twenty-four miles wide. That's when a lot of arguing began.
Nowadays, it's even worse, with nations claiming 'their' watery boundaries end at the end of the 'continental shelf,' and some such. Once again in comes down to money: where the fish stocks lie, where the undersea oil wells are, et cetera.
The obvious answer it that it belongs to no one yet everyone; one of those dichotomies that have to be recognized and dealt with in a civilized fashion to the benefit not only of human beings but to all that reside both on land and in The Deep.