Google chrome still has issues with eID and macOS high Sierre, google is looking into this.
BEIDtoken does its work behind the scenes, so its normal you don't see anything after pressing the Register button.
It is the container app for our BEIDToken service (apple requires a container app so services are no longer installed silently as a part of an installer)
For Safari:
You can check with Applications/Utilties/System Information -> software/Smartcards if the BEIDToken is installed as a smart card driver.
When you enter your eid card and refresh the system information (command R), you should also see your certificates showing up there.
If you certificates are not showing up there, can you check if they are present in Applications/Utilities/Keychain Access?
In that case, macOS 10.13 would have picked up our old tokend, which would no longer be supported by the OS.
Our old BEID.tokend used to be present at /Library/Security/tokend/BEID.tokend ;should you find it there, please remove it and try to authenticate with safari once more
Wkr,
Frederik