Similar solution as Daniel earlier but I copied the URL Firefox (my default browser) was using and opened it in Chromium. After logging in there the url parse voodoo worked and VScode is happily logged in.

OS: Ubuntu 24.04.2 LTS (64-bit)
VsCode version: 1.100.0
Firefox Version 138.0.4 (64-bit)
Chromium version: 136.0.7103.113 (Official Build)