Yes, there are separate processes, however a new instance is not started each time you open it - the same window is re-shown.

    Taskbar: explorer.exe
    Start menu: StartMenuExperienceHost.exe
    Searching: SearchApp.exe

So, when you hit a key with the start menu focused, it closes the start menu and opens the search window (with the search text initialised with that key).

Perhaps keystrokes are lost during the switch.