If you're on Windows 10 try using the Skype Preview from the Windows Store and see if that version works, you might even prefer it to the Desktop version. I am not sure myself why this error is occurring but you can try asking on the Skype forums for assistance.
I think the chances of this being related to browser aspects would be to do with IE or MS Edge as they would use the same engine implementation for accessing the login page within Skype GUI for sure. Any extensions on IE or MS Edge?
Do you have anything like a system-wide ad blocker enabled?