I am Nikon, and don't know any details about what Canon might do, but the shutter count is in the Exif data. Of course, Exif data is not necessarily retained in many online images. And not all of the Exif data is necessarily retained by various image editors (Adobe) when they rewrite the file.
And note too, Shutter count is not in the Exif proper, but is instead in the Manufacturers section. But the manufacturer can and does modify the format of this section from time to time. This is why raw editors need revisions themselves, to match new cameras.
File numbering is meaningless regarding shutter count. We change folders or folder names and reset file numbering. Also movie video use may affect file numbering, but not shutter count. etc, etc.
Most Exif Viewers are several years old, and are essentially abandoned, that is, they have not been modified for years, and they can lose ability to show the manufacturers section, either at all or properly. If your Exif viewer is older than your camera, be very suspicious.
One Exif viewer that does get constant revision attention is ExifTool, so it is easily the most reliable one, day and night, it seems to get a couple of revisions each month. It is command line only, but some notes describing easy Windows GUI use of it are at
Camera Exif data