Most likely the headphones are doing the translation and end up sending a normal signal like any other headphone would. Meaning that if it's not working it's either the app you are using in Linux, the drivers, or the DE.

Give us some more details on what software you were using on Linux, what distro you are using and what DE

Maybe start by checking if other headphones have the same issue and maybe the arch wiki since it has a Bluetooth page

I don't think the device decides if your touch controls work. I'm sending my Sennheiser CX's back because the left bud does no longer respond to touch.

In any event, it seems like you need Bluetooth Avcrp Transport drivers, maybe Linux doesn't come with that.