I found out that by default, Zoom lets attendees change their own names during a meeting at any point. However, sometimes this option seems to be missing. The reason for this is that Zoom has a setting that lets Hosts choose whether they would like to allow the attendees of a meeting to be able to change their name once a meeting has started.
I found this information here:

There are several ways to solve this problem. The easiest way is probably to ask the meeting host to enable the setting, that allows participants to change the name. I solved the problem by leaving the meeting, logging into the Zoom app, changing the name there, and then rejoining the meeting. It would have been much easier to just ask the host to change this setting. Fortunately, I was able to rejoin the meeting fairly soon, so I didn't miss any part of the lecture.

Cannot change my name on Zoom

After joining a Zoom meeting, I noticed that I can't easily change my name. Normally, there is an option to change my name by going to "All Attendees" and hovering over my name. However, for some reason, this option was not available during a lecture.
Subscribe to rename