It's wrong my friend. Real Madrid was not getting him for free. Mbappe left from PSG as a free agent, but Madrid paid 180 million signing bonus to sign him into the squad. In conclusion, Mbappe was not free at all.
But it is a contractual agreement between the two parties only while Real Madrid did not pay a penny to his old club PSG so it can still be said to be a free transfer because if Real Madrid spends money to buy from PSG Real Madrid will spend even more money not to mention the other bonuses so for the contract agreement it is already a matter for the club and the player.