The gray shallot originated in the middle east where it still grows wild. We call them French gray shallots probably for the same reason we call dwarf box "English" box: because that's where our predecessors more than likely got them. French gray shallots are not native to France.
As far as I know, both culinary onions and garlic are known only as cultivated plants; culinary onions are grown from seed, but garlic apparently is not known to produce seed. My French gray shallots have not bloomed here, and I don't know if they produce viable seed or not.