Nero does have a utility which will give you a lot of info about a disc, and it would identify the disc as a DVD-R, or a DVD+R, or a DVD-RW, or a DVD+RW or a video CD, if it were a copy. If it were not, then it would identify it as a DVD.
The program I refer to is called “Nero Info Tool.” It comes with the Nero DVD software suite of programs. It might also be available for download at Nero’s web site, but I am not positive on this.
Also, there are four or five other utilities out there that can do the same thing. If you do a search on Google, I am sure you will come up with some. Perhaps try the keywords “DVD identifier” and “disc identifier” and see what happens.
Now, as to this disc… If a person uses DVDShrink (the popular software program) to rip a file onto their computer, and then uses Nero Burning Rom or DVDDecrypter in order to burn it to a blank DVD, the program defaults are to give that new copy a name of: “DVD_SHRINK” It is easy to change this default, and it is also easy to just use the name of the original film, instead. However, if the user is not aware of this option, or is too lazy to change this, all of their copied discs would then be given the name: "DVD_SHRINK. I have a suspicion that this is what has happened in this case.
In other words, yes, based on what you wrote, I am guessing that this disc is a copy.
For fun, you could also try to copy the files onto a PC using Explorer or My Computer. See if this works, and see if it will then play the files from the hard drive. In other words, treat it as if it were a floppy disc, and see if it will let you copy all of the files. If it does, with no problems, then you know that the copy protection has been removed completely – which would also indicate it is a copy.