"To apply for one, you need to make visit a Police and Border Guard office in Estonia – there you need to submit an application and provide biometrical data (your facial image and fingerprints) for background check."
From the leaflet [PDF] http://e-estonia.com/wp-content/uploads/2014/10/Become-Eston...
Wouldn't this kind of ID be granted to anyone, so long as they are who they say they are, so long as their identity is vetted.
I think selectively doling out benefits or rights is the job of a separate system. (And if everyone has one of these IDs, the design of the second system could be simpler than it might be otherwise. I think.)
I think that our local spy people have access to them (there was a theory about them managing our offshore database backups and therefore never asking for anything); if and how they share it is not disclosed.
foreign governments might get access through travel information agreements (biometric passports were a requirement for free travel to US) but I am not certain about those details.