If you can login to only some of the e-services, it can be caused by the broken session stored in the webpage cache, which happens if instead of the ID card certificate the com.apple.idms.appleid.prd. certificate was chosen.
If that's the case, do the following:
• Open KeyChain Access and make sure that it identifies the ID-card in the smartcard reader.
• Enter com.apple.idms to the search on the right side
If you got positive search results:
• Click on the first, second, third etc search result.
• Check whether any of the retrieved results is related to the e-service that you cannot login to.
• If you find something similar to the below screenshot, delete it:
Then close your browser window by clicking on QUIT and open it again. You should now be able to login to e-services.