Symptoms

I used KeepassXC as my system Keychain quite a long time without any problems. Recently I noticed VSCode skipping keyring unlock prompt. Even if KepassXC database is locked, it somehow stores and remembers login credentials. Other applications, like Slack was more than happy to use DB triggering unlock prompts, whenever DB was locked, hinting that KeepassXC and whole Secret Service thing working correctly.

Fix

Go to $HOME/.vscode, find file called argv.json. Open that file in any editor and search for this line:

{
  "password-store": "basic"
}

Available values are kwallet5, gnome-libsecret, kwallet, basic. Apparently this setting allows you to hint VSCode about keychain you're using, if app fail to detect it for some reason. Remove that line entirely or change value to gnome-libsecret to get correct behavior.
More info from VSCode documentation can be found here.