iPods do not use your computer's user account. Instead, if you use automatic syncing, it is associated with an iTunes library. If a new computer has a different iTunes library, you will not be able to use automatic syncing with the new iTunes library until you change the association (replace the iPod's existing content with content from the new iTunes library). Doing a Restore from the new iTunes library should have changed the association (by erasing the iPod), however... so I don't know why it did not work after three attempts.
In the future, you should move your entire iTunes library from the old computer to the new computer. The process is described in this document
iTunes: How to move your music to a new computer - Apple Support
in the section called "External drive." It is very detailed, but this is what you are doing. On the old computer, consolidate all of your iTunes data in the iTunes folder, which is located in the user home folder's Music folder. If you were using standard settings in iTunes, the iTunes folder may already include all iTunes data. The iTunes folder includes the iTunes library database file (which is what the iPod recognizes as its current "iTunes library") and the iTunes Media folder with all the iTunes media files. Copy that entire iTunes folder to an external drive. On the new Mac, there is an existing iTunes folder. When iTunes is NOT running, replace that existing iTunes folder with your old iTunes folder on the external drive. The next time iTunes runs, iTunes uses your old iTunes folder and everything looks like it did before. And your iPod recognizes the iTunes library on the new Mac as the same library.
Syncs fine now, although the library remains a bit confused.
If your "old hard disk" still has your complete iTunes folder from the previous computer, you could follow the procedure to transfer the entire iTunes folder to the new Mac (not just the song files). You would then have your old iTunes library, including supporting data such as playlists, ratings, and play count, on the new Mac.