Image classification: Predict all objects' classes without bounding boxes
Object detection doesn't mean bounding boxes. It is as the name suggests to detect objects in an image being single or multiple which is what you actually need in your problem.

The display of bounding boxes is your choice if you need the class along with its position or not. As you already get bounding boxes in your solution, you might be getting their classes as well. Simply, don't draw the bounding boxes using OpenCV or any other. Just comment out those lines.