Good question.
Life evolved in the ocean, so the inside of marine species’ cells has a saltiness that is like the saltiness of sea water.

To understand the problems with water and salt balance in living things, you need to understand a process called osmosis. i’m going to start off by giving you a “thought experiment” to imagine. Picture two rooms that have puppy pens in them. The pens are connected by a puppy door through the wall. Puppies can go back and forth between the rooms using the door. Imagine that they are moving randomly. About what fraction of the puppies would be in each pen? Probably about half and half. Now imagine that puppy-loving people are put into ONE of the pens. People won’t fit through the puppy door. Now each person is holding and playing with the puppies, so almost all of the puppies are in one pen. Why am I telling you about puppies when you asked about fish and seaweed? We’ll get there. If you had a cell that contained only water, and put it in pure water, water would enter and leave at about the same rate. The cell wouldn’t gain or lose water overall. This is like the puppies moving back and forth on their own. Cells do not contain only water, though. They contain salts, sugars, proteins, and other stuff. These act like the people in our thought experiment. The salts and such “grab” the water molecules. Osmosis is a process where the water “follows” the salt or similar stuff across a membrane that lets water move, but does not let the salt (and stuff) move across it. Obviously, the water isn’t really going after the salt, but when its random movement brings it to the salt, it is attracted to it by charges and is not likely to leave.

If you dropped a fish’s cell into pure water, water would enter the cell a lot faster than it left because of the salts and stuff in the cell. Eventually, the fish cell would fill with water and burst. If the fish’s cell did not have a lot of salts and stuff in it, and you dropped it in sea water, most of the water would leave the cell, following the salt outside. The fish cell would shrivel up. In salt water, a fish cell has to have enough stuff that is “sticky” for water to keep its water from following the sea water salt out of the cells. Salt water fish have skin that is mostly water-tight, but they do lose some fresh water and take on some salt through their skin. Boney fish can produce a very concentrated urine to get rid of the extra salt. They also secrete extra salt through their gills. Sharks and their relatives have glands that secrete excess salt into their digestive system so that it leaves in their waste. So they don’t separately create fresh water from the sea water, but they can get rid of the extra salt. Seaweed is technically an algae, not a plant, but they deal with water balance the same way. Seaweed can make stuff that is also “sticky” for water and stays in their cells. The salt doesn’t flow into the cells much and the water doesn’t leave much. Algae and plants can get away with having water attracted into their cells because they have a rigid cell wall outside the delicate cell membrane. Think of filling a water balloon. If you put too much in, the balloon will burst like the fish cell in plain water. If you put the water balloon inside a box of the right size, it would just fill the box and stop taking on water. That’s how a cell wall protects a cell from letting too much water in.

What problems do you think freshwater fish have? How might they solve them? You may be interested in studying environmental physiology.

Actually we have a lot of salt in our blood, mostly in the form of sodium ions, which are positive, and negative ions such as chloride ions that neutralize them. There was a physics post doc in my lab once who was frightened that we would die if we drank the pure distilled water in the lab, because red blood cells swell and burst when they're put in distilled water. He was wrong, and it was quite a stupid error about biology. All living cells have pumps in their outer membranes that pump ions in and out of the cells, to give the right amount of salt in the cells and in the blood around them. The pumps also use chemical energy to pump potassium ions into the cells and sodium ions out of the cells.

That sort of explains your question about how plants and animals live in salt water. It doesn't seem any more strange to me than how we live in air, or how other plants and animals live in fresh water. We and they all have the pumps and other complex molecules needed to keep the right amounts of salts and water in our bodies.

I have an idea about how life started between sheets of mica. This idea explains why living cells contain such large amounts of potassium ions, because potassium ions hold mica sheets together. You can find out more about this idea by googling : granny mica sheets.

The important thing to realize is that what’s important for an organism is water and even the saltiest of seawater is still about 96% water. But yes, the remaining salt could be a problem for a sea plant or animal. The important thing is that organisms have ways to use energy to pump salts out or sequester them away from the important cellular machinery. With plants, they can store salts or excrete the salts through leaves. With saltwater fish, they deal with the high concentration of salt by having urine that has a high salt content. The basic idea is that living things can adapt to salty environments in various ways by either hiding the salt away in a separate cellular compartment or excreting the salt.

The short answer is that seaweeds and other saltwater algae aren't actually plants, and use internal biochemistry that does not require fresh water the way that land plants do. That said, seagrass IS a plant, but it, too, has managed to adopt biochemistry that can make do with salt water instead of fresh water.

Plants and animals that live in saltwater have a few mechanisms to regulate their physiological salt content. This referred to as osmoregulation. Osmosis refers to the flow of solutes from high concentration (salty; high osmotic pressure) to low concentration (dilute; low osmotic pressure).

Many saltwater fish osmoregulate by active excretion of salt through the gills and salty urine. This allows the fish to maintain lower salt concentration in its body than in the surrounding saltwater. The adjective "active" means that the process consumes energy. The fish's body is pushing salt ions opposite the direction that they would naturally diffuse by osmosis, and this consumes energy, which the fish has to provide through its food consumption.

Seaweeds are a bit more complicated. Many seaweeds synthesize alcohols and other organic molecular solutes inside of their cells in response to increasing salinity levels from their environment. This increases the osmotic pressure inside the cell to match the outside environment and thus prevents excess salt from diffusing into the cell.