Is adding the word "al-Madani" to one's name after graduating from the Islamic University in Madinah considered a correct change to the name? And is there any record of any of the Companions or the Successors adding the name of their city of study to their name after completing their studies there?

It is permissible for someone who resides in a country other than their original homeland to be attributed to it, even if they were not born there or if it was not the home of their fathers and grandfathers. This attribution does not negate their origin from their original homeland. This is what scholars traditionally practiced, and an example of this is: "So-and-so al-Basri then al-Makki," meaning his origin is from Basra, and he resided in Mecca. Many examples have been cited of scholars who were attributed to places where they settled and resided for the pursuit of knowledge or other reasons.