The LEDs look like MBI Matchlights with the rubber sheath cut off. The link is to an old Indiegogo campaign for an emergency LED that lasts for 8 hours and is then disposed of. I don't know if the design is proprietary or if they're assembled from existing parts. Unfortunately it looks as though they were a one time offering, but maybe someone who knows more about this stuff than I do has a better idea.
I've done further research, and I think I've narrowed it down to either a cr322 or a cr435 lithium pin battery. I still know next to nothing about any of this, so if someone has anything to add, I'd be more than delighted.
Yeah I think you're onto something with those batteries. Further googling reveals they're used in several products eg fishing tackle and archery nocks. They're probably used in all sorts of stuff. Very cool little gimzo, I've never seen anything like it before.