In practice, countries recognize only eID cards they produced and signed themselves (well, ordered from Gemalto, Giesecke and Devrient or IDEMIA).
Now, once the use of eID cards expands, all these use cases will require software. So far, countries do on their own. Belgium writes its own software, Estonia as well. The point is that most of that software could be reused.
So far, only open source parts are reused, like opensc.
The Estonian e-signature software is fairly well written, open source and easily extensible to other countries eIDs. Latvian and Lithuanian are already supported.
Now, once the use of eID cards expands, all these use cases will require software. So far, countries do on their own. Belgium writes its own software, Estonia as well. The point is that most of that software could be reused.
So far, only open source parts are reused, like opensc.