I'm going to dump a lot about Antarctica right here. It's a place I love.
Seasonal ice is due to changes in the Earth's tilt and the drop in temperatures. What freezes is the ocean, not fresh water on the Antarctic Continent. Antarctica is a desert, so what you're seeing is the little bit of snow that's built up over hundreds of thousands of years*. It doesn't get warm enough for any of that ice or snow to melt, so there is no fresh water melt to freeze in winter.
If you hear something referred to as an "ice shelf" that means that the ice and snow (not saline) have been around for a long, long time. Snow accumulation and glacial activity (perpetually frozen fresh water rivers) have extended these ice shelves out into the ocean - cantilevered out and/or suspended at multiple points.
What makes this so different for Antarctica from the Arctic is there is no continent, no land at all for the Arctic. In the Arctic there really aren't any ice shelves. If you ever hear someone talk about Arctic ice shelves, they're going to be talking about ice shelves extending from Canada's most northern land mass, or Greenland. Extremely small in comparison to the Antarctic.
Antarctica is also the highest continent on Earth. These massive ice shelves are attached to and supported by the continent itself, and by islands and other land masses. When these ice shelved break, they drop. The oceans rise. That does not happen in the Arctic**. The Arctic is like ice cubes in a glass. The water level doesn't change as they melt.
* True for the main continent, the peninsula extends into warmer water and further out, and accumulates snow and melts seasonally with summer temperatures sometimes reaching ~60F. The warmest parts of main Antarctic Continent have not historically gone above freezing.
**Arctic in comparison to Antarctic is the ice cap extending from the poles, and does not include Greenland for the Arctic, for obvious reasons.