It's a requirement to prevent mixing up classified and unclassified information. Computer screens display banners like thisWould recording the screens inside the CIC/ICC happen often enough to warrant including that as a feature though? Most Navy members who have commented on the video have remarked on how unusual it would be to have a phone out recording anything inside that area. I think the "target not identified" idea makes a little more sense given the context.
Displaying "target not identified" would make more sense than "UNCLASSIFIED" in green.
Jeremy Corbell stressed that the video wasn't illegally recorded on a cell phone, but that a "VIPER team" was invited to record it, and it's unclassified.