Workaround
Copy the widevine library from Google’s Chrome and restart Brave:
$ cp /opt/google/chrome/libwidevinecdm.so ~/.config/BraveSoftware/Brave-Browser/WidevineCdm/
This upgrades “Brave`s” widevine extension from version 4.10.1196.02 to 4.10.1440.182. The versions are visible in the payload of the POST request to https://gew-spclient.spotify.com/widevine-license/v1/audio/license.
By using Google’s extension Spotify’s web player works as intended again. Successfully checked that it doesn’t break Netflix in the process.
Update:
You can obtain the working library also from Google’s servers:
https://dl.google.com/widevine-cdm/4.10.1440.18-linux-x64.zip