Recent posts widget : how to display only the posts of the current category?

Use the ‘widget_posts_args’ filter. Callback code could get the current post’s category (for singular posts), then add something like
'cat' => $category_id,
to the passed args array.

This worked for the legacy widget. I’m unsure if it still does for block based widgets. I think it does, but I’ve not verified.

@bcworkz

Do you have the full code for that?

It seems the right solution for my “wish-list” ??

Something like this:

add_filter('widget_posts_args', 'my_restrict_cat', 10, 2 );
function my_restrict_cat( $args, $instance ) {
  if ( is_single()) {
    $cats = wp_get_post_categories( get_queried_object_id(), ['fields' => 'ids',]);
    $args['cat'] = $cats[0];
  }
  return $args;
}

Untested though. Only restricts widget queries on single post pages. Uses the first category term in the returned list regardless of how many are assigned. You can add this to your theme’s functions.php, but if the theme is subject to periodic updates it should be in a simple custom plugin instead. Plugins aren’t all that difficult to create.

$args is passed directly to a new WP_Query object, so you can alter the query in any other way you like as well.