I'm pretty new to Blender. I added Blenrig 6 specifically to use its picker for character animation (the Spring rig goes with it). I wanted to use the picker for other rigs, like Ellie on Blender Studio, but nothing shows up in the picker window when I select Ellie. Is there some way to create my own picker set for that rig, so I can select her bones from it? Is there some other tool that would be better for this if I'm only doing character animation and not rigging?
I don't see mentioned Blenrig's picker being able to work with other rigs, not at a glance at least and it's fairly common for rig generators (their goal is to make a rig and its tools after all, they rarely support anything other than themselves).
Generally speaking, independent pickers are made manually by someone either fully hard-coded manually, or using a third party tool specifically for that. Unless someone made a picker and made it available, and you have that same third party tool if required, you won't have anything.
Back in the old 2.7 days, there was ChristopheSeux/rigUI: Blender addon for making UI for rig. It hasn't been updated for the past six years, so I bet it doesn't work in modern Blender, but perhaps someone forked it somewhere and made it work.