I have a MacBook Pro. I pay monthly for family Apple Music. I have just purchased an iPod shuffle only to discover they aren’t compatible. My old iPad works and all the other devices but not this shuffle.

And one lead me to an easy free solution please

DRM protected stuff (all music on Apple Music subs) won’t work with a shuffle or a nano. I’m in the same boat.

That’s unbelievable. I can download it to my ancient iPad but not this apple device.

The iPad has a connection to the internet that can be used to verify your Apple Music account. The Shuffle does not have any way to connect to your account or the internet.

That’s unbelievable.

Its not unbelievable – its the nature of the service you’re consuming. You can put other music on there – music you’ve bought or imported from CD etc – just not music you’re renting. Its the wrong tool for that job.

Apple Music is a streaming service, that’s all it is. If you want music on a portable device like a Shuffle, you need to rip a bunch of CD’s or download tracks from the Music store. Using Apple Music means you’re just renting the music, you don’t own I.
I won’t use any streaming service, I either download occasional tracks, or more often buy CD’s and rip them into iTunes, or Music as it’s now called in OSX Catalina.
I spent £45 on CD’s in HMV Bath today, a 4-disc Kate Bush collection, and new albums from Thea Gilmore and Big Thief; they’ll be ripped and put on my phone shortly.