This was his main symbol because it was his weapon of choice and he used it to control water. The Cyclopes made it for him before the war of gods and titans.
The horse is a really common symbol of Poseidon because he created the horse and is known well for it. There are also many variations of the horse that he made. Some examples are the Pegasus, unicorn and hippocampi.
This is one of his symbols because the Dolphin is a sea creature and he is the Sea god. It was said that Dionysus created the dolphin when torturing some sailors.
Some say that this was one of his symbols because it was accidentally created while Poseidon was trying to make a horse. Others say that it was because he sent a bull to king Minos of Crete to sacrifice.
He was the god of earthquakes, of course. He was given most of the responsibly of earth after Zeus, Poseidon and Hades drew lots. So technically, he rules the earth too.
He had this symbol because he controlled the waves and he got this power by choosing to rule over the sea.