Yes, there is exif tags on videos including .mov files, there should be some apps that permit you to view such tags, but they maybe incompetent, you could try converting the video to another format, most of the times the conversion does not take the exif tag with it.

Sorry that I could not bring you a solution, but you could try with any video editing software, but if after all you are still concerned, I suggest not to upload the video to the interwebs and store it on a portable drive or any storage that does not have any access to the internet, since keeping it on the phone might be a risk, because of all of the social media apps ( Facebook ( and its messenger ), G+, Twitter, Instagram,etc ) most apps request access to the gallery,records, etc.

Best of luck