Yes, the queen herself has seen The Crown (though Prince Philip apparently isn't a fan), but that's not the only royal show she's into. Her Majesty reportedly also enjoys Downton Abbey and specifically looks for historical inaccuracies while viewing.
But just because she's a serious monarch doesn't mean she can't enjoy a little mindless TV — the queen was apparently into The X-Factor and had her staff record it so she could watch the following day.