It's not a problem with the iTunes version, or even an actual problem (as in a "bug"). It sounds like it is working as designed. The 2nd (and 1st) gen shuffles can only be associated with one iTunes library at a time. Since you are connecting your shuffle to a new computer, it has a different iTunes library from the one the shuffle is currently set up to sync with.
What you need to do is have iTunes on the new computer sync your shuffle from its iTunes library. This will replace the shuffle's existing content. Whatever is on the shuffle now that is not in the new iTunes library will be lost.
If you still have access to your old computer, ideally, you should transfer your complete iTunes library from that computer to your new one. Then, the shuffle will treat the iTunes library on the new computer like the old one. Or, if you have a backup of the old iTunes library, add those songs to your new iTunes library.
If that is not possible and your only copy of the song files is on that shuffle, you'll need to off-load the song files from the shuffle. You cannot use iTunes to transfer song files from iPod to computer, except for songs purchased from the iTunes Store. However, there are third-party methods and utilities that can transfer from iPod to computer. If you do an Internet search on something like "ipod transfer," you should get some links. (The CNET article seems to be popular.)
Once the song files from the shuffle are on your computer's drive, add them to your new iTunes library. Then, you can sync the shuffle from that iTunes library, as desired.