Yes, with special Electron shenanigans that still work in a normal browser well I suppose. The web UI has a few features that make it more device agnostic too (e.g. a hamburger menu to replace the File menu. Yes, you can run Codespaces on an iPad.)

Code server (https://github.com/cdr/code-server) works in a similar way using your own PC as a host, but for whatever reason Codespaces is more performant.