Having seen about 30 polissoir stones in France, I have to say that this does not look like any of them, the grooves are much narrower, and nowhere near as smooth. However, the combination of grooves and large rounded basins is very common, grooves for the edges and basins for the sides...

As for the type of stone, the French ones come in all shapes and sizes, and all sorts of stone. I have seen them in quartz, sarsen type stuff, and hard flinty lumps found in a chalky limestone region, but I am no expert on stone type.

One thing that is normally a must is the presence of water, used in the polishing process. Most I saw were near a stream or river. Unless the stone has been moved to its current position at some later date, I would think it unlikely to have been used for polishing where it is now, as it looks like there is no water source.

I could of course be completely wrong.