Praise be to Allah.
The chat history or messages that reach someone belong to him, and the sender has no right to them. But if the sender feels that some harm or negative consequences may result from keeping these messages, because there is a possibility that others may read them, and he asks you to delete them, then you should respond to his request, so as to ward off harm from your fellow believer, or you should let him know that you are going to keep them in a safe place.
Whatever the case, you do not have to respond to his request, and he can delete the messages from his own device if he wants, but he does not have any authority over the device of someone else.
Once a word is uttered by a person it controls him, after he was controlling it.
Ash-Shaafa‘i said to ar-Rabee‘ ibn Sulaymaan: O Rabee‘, do not speak about that which does not concern you, for if you speak a word it controls you, and you no longer control it.
But for the sake of brotherhood and maintaining friendship, the individual should respond to his brothers’ request, when there is some benefit for them in that and it does not harm him.
And Allah knows best.