I'm not quite sure what you're after, but:
$ ls *.jpg
$ jhead -n"%Y%m%d%H%M-%f-renamed" foo.jpg
foo.jpg --> 201909141339-foo-renamed.jpg
$ ls *.jpg
The usage documentation for jhead is quite good, and you might want to check the manual for strftime (if you're on a Unixoid system, which it looks like you are), which ...
You could use FFMpeg to strip the video stream from the MOV files. You could then follow up with exiftool to try and copy any metadata that FFMpeg didn't copy.
First run FFMpeg
ffmpeg -i Input.Mov -map_metadata -1 -vn -c:a copy Output.mov
This will copy the audio stream to a new .MOV file and drop the video stream.
Then run exiftool
Take a quick photo of your feet before taking your photos. If it's quick and easy, make it low res, low quality to save space in memory, of course switching immediately back to high res high quality.
"Those aren't my shoes!"
A Canon-based answer: You can add a 1-5 star EXIF compatible rating to each photo (there is also an option to set it „star“ or „no star“) with a button. One of you could set such a rating after each photo.
Only a passive or natural solution will beat taking a selfie/picture of your hand first. It's a shame the NFC feature can't be used to identify the hand holding the camera. It can still do some good.
Tapping the camera with an NFC-enabled smartphone after shooting would work by getting the picture onto the photographers phone (according to the manual; ...
Many cameras can add Author and Copyright to the Exif. When switching cameras, you can edit the Author. This would be cumbersome without a touchscreen.
Consider taking selfies or pictures of ID badges, as flawr suggests.
The easiest way, this is guessing you have a digital camera, I caught something about an SD card. THE EASY WAY IS TO RESET PICTURE COUNT BACK TO OOO!; each time you finnish, asking the next user to do the same and show them how. So all pictures are in blocks of numbers, starting from the fist user, ending with the last user. You will find the option in the ...
Add value to your data.
Attach a tiny clap board to the camera strap with a white board marker. Write the lighting conditions, location (including studio, expo stand or office details that are not available from GPS), event if you moved back and forth between more than one on a given day, photographer and photo model details on the little white board and ...
You've been offered some ingenious methods, but you're not really going to go to the trouble of any of them, are you! Either buy two cameras, or convince yourself it doesn't matter which one of you pressed the button.
One solution that might work if you don't switch too frequently:
Take a selfie whenever you take the camera.
Then you know all following pictures have been made by the person of the most recent selfie. (Maybe you should think about a "sign" if you do frequently take pictures of eachother.)
I did this at a previous job, where we first also used to keep a ...
Nikon D3400 (and, I assume, other models) lets you select the active folder to store files in. Just change folders when you change photographers.
More generally, you can use two memory cards and change cards when you change photographers.
There isn't really any surefire way, other than meticulous bookkeeping, or following consistent habits.
Use your mobile phone to take images of the rear LCD info page showing the file name for the first and last image each of you take each time you operate the camera. For instance, if you take a dozen pictures, when you're done shooting for a ...