I created a navigation link Class photos

Within that I created a photo gallery Upload your photos here

My classmates want to upload a set of photos for a specific topic (class reunion 2020) to put their photos in. When they upload there is no way for them (that we have found) for them to upload a set of photos and give them a title. It only shows up as their name. When they then upload more (class reunion 2015) they still go into that same one with their name.

Can a classmate without admin priv do something like this?

I couldn't locate that specific gallery to check... go to your gallery and be sure the upload permission are set to All Classmates. This will provide a button at the bottom of the page to start a gallery when they arrive to that page.