Take some close up photos of the handle details. Not something I recognize but more detail could help someone ID this faucet.
Often times you can replace the seats, and those are often generic if they are replacable at all. We just got rid of almost all of our old seats and washer kits since repair of old faucets is so uncommon. But replacing the stem isn't going to repair the seat which is what is causing premature failure of the washer. You can usually look down the opening and see if there is a square/hex shape inside the seat for removal. Or even ReDress the seat if the gouge isnt' too deep.
