You should be able to wipe the iPod and start again, even though you're not using the same computer (and iTunes) as before.
I think first of all, you should ensure that the iPod really is an iPod Classic, as opposed to an iPod Touch. The significant difference is that an iPod Classic is fully dependant upon (a desktop) iTunes and neither the original computer nor his name should prevent a change of media on it, but an iPod Touch can access the iTunes Store independently of iTunes and can have items on it that are tied to a specific account.
Check this page to confirm which iPod you have: Identify your iPod model - Apple Support.
This is an iPod Classic:

Assuming for now that the iPod is a Classic, simply connect it to your iTunes and attempt a Sync. During the process you will see a message telling you that if you sync the iPod with your Library, all the existing content will be removed and iTunes will add to the iPod what is in your Library. If you agree to that, you will also need to set specific Sync options to let iTunes know what content (in its Library) should be added to the iPod.
Using the screenshots below, try the following steps once the iPod is seen by iTunes:

- this will display the Device/Settings/Summary pane
- switch to the Device/Settings/Music pane (1.)

- tap the Sync Music button (2.) to ensure there is a tick in the box (a checkmark)
- at this time, tick the Entire music library box (3.). That will add all the music in your library to the iPod. Note that you will not see any of the sections numbered 4.
- tap the Apply (or Sync) button in the lower right corner of the iTunes window (not shown in the screenshots)
- once the Sync completes, the music in your iTunes Library should be on the iPod
If the procedure does not work as expected, please post back with clear information about which step didn't work and what did happen. Include any error messages that you see.