This soft stone was formed in ancient seabeds made of clay.
Broken slate tile.
To replace a damaged tile you simply scrape out the grout around the tile break up and chip out the tile then scrape off the old adhesive which most likely is thinset a grout like mortar adhesive so you can start over with a clean subfloor surface.
Home with slate roofing system.
Scrape out as much grout as you can.
Drag the narrow flat edge of a small pry bar or a small chisel or along the grout lines surrounding the damaged or lifted slate.
This includes color and texture or anything that you like.
Very few types of roofing materials can match the natural beauty and longevity of a natural slate roof.