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.

Signing in the browser uses a extension, code here https://github.com/open-eid/chrome-token-signing