It obviously believes that the iPod Nano was last synchronized with a library on "John's MacBook Pro" – or it would not be showing that message.
If that name is coming from flash storage somewhere on the iPad Nano (as seems likely), and it's not correct, that raises the question of whether the iPod Nano is experiencing some sort of hardware or software failure that keeps it from consistently remembering its connection to your new computer.
One thing that comes to mind – even though it seems like a remote possibility: You can only rewrite a sector of flash memory so many times before it stops accepting new writes. Sometimes when a sector of flash memory "burns out" (so to speak), you can still read the old contents; you just can't change them.
Could this have happened to some of the flash memory on your iPod Nano? If it is unable to accept new writes, then there might not be a good way to change the song selection on the iPod … even if you could keep charging the iPod again and again to play the current selection.
You could try restoring the iPod to factory settings. If the problem is due to a software glitch, a reset followed by a reload might fix the problem. On the other hand, if the flash in the iPod Nano is "burning out", a reset might be just the thing to brick the iPod for good.
Restore your iPhone, iPad, or iPod to factory settings - Apple Support
I don't believe Apple offers hardware service for iPod Nanos any more.