Apple Music is not "purchased music". Instead, Apple Music is a subscription service which gives access to Apple's music library while the subscription is paid.
I believe that Apple Music requires iOS 8.4 or later on the iPod itself in order to work. So if it's an old iPod Touch (for example the 4th generation or earlier), you cannot add Apple Music to those models as they cannot use the minimum iOS required, which is iOS 9.3.5.
The oldest iPod Touch that can use Apple Music will be the 5th generation one, which can use iOS 9.3.5.
You can find out which generation the iPod is by checking which version of iOS is on the iPod. To do that, look on the iPod under Settings and then, on a 2nd gen model for example, it's General/About>Version (so that's Settings/General/About>Version). On other models, the menu structure will be very similar, if not the same.
In the meantime, the old iPod can still store and play music that you have purchased, either from the iTunes Store, other online retailers or copied from CDs. However, that feature and Apple Music do not get along - it's one or the other. I'm not sure how you would have Apple Music on one iDevice and not Apple Music (i.e. your library) on the other, especially if (as I assume) the purchases from the iTunes Store were on the same account that now subscribes to Apple Music.
It would be possible to use the old iPod with an iTunes library of purchased music if the iTunes programme was signed into another Apple account. In theory then, you would sign out of the account that uses Apple Music and then sign into the account that does not (or sue a different computer). However, iTunes Store purchases would not be available to the second account if they were purchases on the first account.