Is there a way to see the photo's image name?

If you mean by "Image Name" the filename of the image file, you cannot see it in Photos at www.icloud.com . You can see the title of a photo below the thumbnails or as a caption above the photo. If you are having a Mac, write the filename into the title of a photo on your Mac, plus any additional information you want to be able to see at www.icloud.com.

For example - the photo title displayed above the photo viewed enlarged:

The user interface for viewing your Photos Library at www.icloud.com is very limited. It looks to me like an emergency access to our photos, when we are finding us without a Mac or iPhone or iPad, and have to borrow the device of someone else, to be able to access our photos.

If you are using the Photos App on an iOS device, tap on the (i) icon at the bottom to get the image details. The file name ir right under the creation date.

Someone will no doubt correct me If I'm working, but if using the Photos app on the iCloud.com website I don't believe there is a way to view the file name outside of actually downloading the file.