The extension settings mention windows as example OS name (at "Target platform (windows, linux, darwin)" caption), but this is not correct, using windows will make completion fail.
One should use OS names consistent with FPC (and thus Lazarus, and Castle Game Engine too). Like win64 or win32.

P.S. I'm using your VS Code plugin with CGE LSP server, https://castle-engine.io/vscode . I documented there the need to use win64, not windows, and confirmed it by testing.