Homogeneous tile is usually porcelain clay tile.
Whereas, compressed marble is marble mixed with white cement, about the same as broken marble. This is popular in the 80's. If it's green colour, it is very nice.
Homogeneous tile cannot be polished on site.
If your friend wants to save cost, then go ahead and repolished it. It costs about S$1.20 - $1.50 per sq ft. But the house need to be furniture-free and there's a risk of water leakage to the lower unit.
Contractor will want to advise to hack, it means more profit for him.