The builders did not use caulk in the shower; all the corners and transitions are grouted.
I lifted up the square plate that frames the drain and found a Ã‚¼-inch layer of grout under that, too. It was cracked so I cleaned it away, down to the mortar.
We have another shower and its plate covers a layer of plumber’s putty. Now I don’t know what is supposed to fill in the space under the plate. Plumber’s putty, mortar, or grout?
Doesn't really matter. Unless you want to level it, the cover doesn't need to be watertight or fixed in place, so you can leave it loose -- helps in cleaning the drain if needed later as well.
The water seal is below the level of the drain cover -- it's at the level of the shower liner, usually an inch or so below what you see.
Some folks (even some recognized experts) prefer grout at the corners. For me, it depends on what's behind the tile.