Your compressed images work in the exact same way as uncompressed images in WordPress. You can add alt text in the exact same way, ie. through the Attachment Details editor when clicking on an image in your Media library.
If you configured all your Alt Text for images in WordPress itself, then the plugin shouldn’t change any of that.
All of the Alt Text that you edit in WordPress is saved to the WordPress database, and the Compress JPEG & PNG images plugin doesn’t touch that when compressing images.
I’ve just verified that by disabling the plugin, adding one new image, and editing the alt text of the image in the WordPress Attachment Details editor. After that, I enabled the plugin and compressed the image. After compression the alt text was still there. Both in the WordPress editor and on the page itself.
In your latest screenshot I can see that there is an alt text configured for that specific image. Are you experiencing that the values for the alt text in the WordPress editor were gone after compressing your images, or that they are there, but the SEM rush report is not finding them?