Is it possible to remove the hard disk from a PS4 and insert it into another and continue using all the saved games and data as if it was in the original PS4?
A hard drive used on one PS4 cannot be used on another PS4 unless you format the drive first. This is a security measure put in place to prevent data from being used on anything other than the original console.
The hard drive itself can be used on another console, but you won't be able to use the data or games stored on it. It will need to be reset / formatted.