Some of the story of this style and this server can be seen in this previous thread.
With regard to the style, you can of course fork it and host it yourself - the cost of hosting will vary from a couple of coffees a month for a map of a small area to perhaps a couple a day for the whole world.
The tile server of the layer that you see at osm.org is hosted by OpenStreetMap France. I don’t know what the long-term future of that is - I’d suggest privately contacting the person running the servers (prominent in that previous thread).