Is it correct to analogize the permissibility of studying at a private university that deals with usury to the prohibition of working in usurious banks? And what is the difference between the two cases?

What you told your friend about the prohibition of working in usurious banks, even in security, is the opinion of many contemporary scholars. A fatwa from the Permanent Committee for Issuing Fatwas states that it is not permissible for a Muslim to work in banks that deal with usury in any form of cooperation, such as writing, witnessing, security, and so on, because it involves aiding in sin and transgression. Sheikh Ibn Uthaymeen, may Allah have mercy on him, was asked about the ruling on working in a usurious institution as a driver or security guard. He replied that it is not permissible because a person's entry into a job in usurious institutions necessitates contentment with them. As for your friend's argument that the university accepts money from banks, it is incorrect because giving money to the university does not involve direct or intended assistance in dealing with usury. Some scholars have held the view that it is permissible to work in usurious banks in departments that do not directly engage in usury, such as customer services in trading aspects, and similar jobs that are not directly connected to usury. The proponents of this view argue that these areas do not directly engage in usury, even if they may serve it. However, the preponderant opinion is that it is not permissible to work in usurious banks in any sector whatsoever, out of precaution for one's earnings, and because the Messenger of Allah, peace and blessings be upon him, cursed the one who consumes usury, the one who feeds it, its two witnesses, and its scribe.