Fish used in sushi is almost inevitably frozen, or you’d have a mouthful of extremely rubbery, almost crunchy fish. Remember also that the original sushi was fish fermented in salted rice for a while. As the availability and taste for fresher fish grew, along with modern refrigeration it was discovered that you get a better texture with fish that had been frozen immediately after slaughter.
This, it is important to note, is only really the case with large, slow-growing pelagic fish like tuna.
Freezing also kills some potentially zoonotic parasites that may be present in fish.
Unless you're buying it straight off the docks, or pulling it out of the water yourself, any unfrozen fish you see has likely been thawed on site, after delivery.
"Fresh" fish means that it was frozen on the boat, and just thawed today.
I sometimes make nigiri sushi by sawing slices off of a frozen fish filet with a kitchen-use-only coping saw. Then I put it in the fridge for the next day. Comes out fine. If I wanted, I could freeze the rice with it, and have sushi later in the week.
I have had sushi that was completely fresh — literally a tuna I caught myself, carried off the boat, cut up and ate. The taste and texture was fantastic.
You are right that almost all fish you eat has been frozen. This is partially for practical reasons: fishing boats are often at sea for weeks at a time. Also for safety reasons: most fish have a significant risk of parasites which are killed by deep freezing.
Tuna is one of the few fish safe enough to eat fresh and raw (and even then there is some amount of risk).
This, it is important to note, is only really the case with large, slow-growing pelagic fish like tuna.