I can't rightly say that I am able to navigate the maze of standards and acronyms associated with smart cards, but the OpenSC tools on Linux have worked for me with a couple different smart cards (Nitrokey HSM and Taglio PIVKey). There are quirks. The Taglio PIVKey can't load certificates using OpenSC, but I've always generated the certificates on the device anyway.
Just because there is smart card software for Linux it doesn't mean it will work with $SOME_GOVERNMENT's interfaces.
Ofc in this case feldrim above pointed us to the mac/linux/etc downloads so the estonian government has actually heard there are other platforms besides Windows.
Presumably you'll be using it with a browser. I'm sorry that I didn't clarify that assumption in my first response.
I don't know about Estonia in particular but I'm guessing "$SOME_GOVERNMENT's interfaces" for most places is going to be HTTPS.
So, with that in mind, I've used a Nitrokey HSM and a Taglio PIVKey with Firefox on Linux using the OpenSC tools PKCS 11 module. I would suspect any smart card supported by OpenSC will work fine in Firefox.
From my reading, OpenSC is being distributed by the government of Estonia, so I suspect using it in a browser that supports PKCS 11 modules compatible with OpenSC on a Linux PC would work fine.
Yes, I use the eID software on Linux all the time. It is based on OpenSC and the main stuff in the browser is all standardised. OpenSC is loaded as a plugin to say Firefox, and most of the authentication is standard TLS client cert stuff.
The app is used for changing PINs and there's another one for signing documents.