That floor looks like it might be slate. Of course there are tile products that look like slate. I like slate floors but that one doesn't seem to go with anything else in the kitchen. It's like it wandered in from some other place.
I don't mind tile in bathrooms, but don't want it in a kitchen. Ms Popper does not like cold floors, so we have never installed any tile anywhere. We have tile on our kitchen floor at the moment, since it was there when we bought the place and we haven't overhauled the kitchen yet. We put rugs over it, also not ideal for a kitchen floor but at least it's not so cold and our old dog can navigate safely. We live in a cold place in the Rockies, but then I have not put any socks on yet today and it's snowing outside. I'm sitting around with bare feet, complaining about the cold.