Mbappe will change teams either this season or next season but his team is sure to change. Mbappé would have joined Real Madrid long ago but he is only waiting to become a free agent member but his free agent membership is never possible because PSG keep him in the team with a huge annual salary so when he wants to leave the team, PSG will definitely want something through him.
Mbappe helped PSG to understand that they can not succeed by giving too big power to a player. They failed with Mbappe and they learned a big lesson from him so they will not repeat it in future. I think the cost is paid by PSG but they did also had a precious lesson from money they paid and from Mbappe.
Together with a successful season from Manchester City with an opposite approach than PSG, it will motivate PSG to change themselves starting from this summer by selling Mbappe. What club will be a next club for Mbappe is not important with PSG, they need to get money from Mbappe transfer to buy new players.
A few days ago there were rumors about Mbappe joining Liverpool but I don't think this rumor will be true because Mbappe was getting enough opportunities in PSG but still the main reason for his change is to achieve something which is possible in Real Madrid. If Mbappe were to switch teams, the most likely move would be to join Real Madrid.
Klopp and Liverpool had problems to solve conflicts between Mo Salah and Mane. Will they buy Mbappe and face with a similar conflict between Mbappe and Mo Salah? I believe if Mbappe joins Liverpool, he will not want to be considered as a second brightest start in Anfield. Liverpool don't have money to buy him and they will not bet their club finance balance to buy Mbappe.