I was trying to figure this out all afternoon as well and figured out as well that the className is passed along by default in the save function for the root of
certain elements.
e.g. in the following code example where className is warning
return (
<div className="other-class-name" value={ props.attributes.content } />;
will return an element on the front end of
<div class="wp-block-block-name is-warning other-class-name">
https://developer.wordpress.org/block-editor/developers/block-api/block-registration/#supports-optional has some more information