But how do they put code tags in there? Last time I checked, Markdown doesn't allow Markdown inside HTML tags. Or do they just wrap it in <code><pre> and the syntax highlighter picks it up later in the chain?

Markdown inside HTML tags is fine as long as there's a paragraph break (two newlines) between the Markdown and the HTML tag. So it usually goes like:

    <detail>
    <summary>...</summary>

    Some `markdown`
    
    </detail>

I would imagine this sort of behavior is in commonmark, but I am too lazy to check.