For humans, our blood and other tissues have a salinity which is less than that of sea water. There is an old chestnut that our blood has a comparable salinity to sea water - its just a classic example of textbook error.
The major reason we need to drink and excrete (sweat, breath and urine) water is to rid the system of waste- i.e. water soluble chemicals or heat we do not need. The kidneys do a lot of this work and they need a reasonable osmotic potential to help them.
The osmotic potential is the tendency of molecules, for instance salts, to diffuse from one reservoir of liquid to another. The kidneys burn metabolic energy (food) to create a slightly higher concentration of waste chemicals into the bladder/urine. Other organs will fail or also work to maintain a local salt concentration (e.g. any nerve cell which depends on Na+/K+ concentrations to transmit their signals. But the kidneys are the gatekeepers of salinity in the body. They work hardest (and can get injured) when there is more salt in the water we drink. If you drink sea water you will quickly find your kidneys overworked and not really functioning.
That's why salt water will tend to make you thirsty (if you can drink any at all). Most people will gag when they drink even a little sea water, much less a glassful.
My understanding is that Sea mammals actually eat their fresh water - relying on the fluid they get from fish to keep their salt balance. Its true that some animals like fish and molluscs etc have adapted other means of keeping their salt balance in check. Its just also true that its a lot of work to keep this ability and when animals move away from the oceans, they quickly lose that ability.