No, the URL with its path-(part) is absolute and that's intended. But, when your testing forum shares the database with your live forum (as you stated before), then it shares also the settings table with the settings for the live system (including the URL with the path to the live forum). Thatswhy your testing forum loads the CSS-file (status 2.4.21) from the live forum into your testing installation (status 2.4.23).

Thanks...!

I think that it's easier to create a new database for testing the new version then...

It's a good idea to mention it in the README.md file of a new version:
"To test only the new version you should create a new database..." or something like this (my English is terrible).

Thank you, shian.