Return all Tags and Categories in Separate List

how I can get the categories from the same global search

You can add a taxonomy attribute to your shortcode and then pass the value to get_query_terms_list(), like so:

add_shortcode( 'bd_terms_list', 'bd_terms_list_custom_callback');
function bd_terms_list_custom_callback( $args ){
    $atts = shortcode_atts( array( // list of default attributes
        'taxonomy' => 'gd_place_tags',
        'sep'      => '', // optional, but you might use it someday? :)
    ), $args );

    return get_query_terms_list( $atts['taxonomy'], $atts['sep'] );
}

See the Shortcodes API on Codex for more details about handling shortcode attributes.

And for example with the default post_tag and category taxonomies included in WordPress core, you can display the tags and categories in separate list like so:

Tags: [bd_terms_list taxonomy="post_tag"]

Categories: [bd_terms_list taxonomy="category"]

So you really just need to use the correct taxonomy slug, just like you could see above.

In response to your comment:

the slug for category is ‘gd_postcategory’ without the underscore for
some reason. I’m not sure how to fix the underscore, so I added 2 x
shortcodes now instead of using the more elegant taxonomy=”category”.
It works 🙂

Why would you need to “fix the underscore”?

And you do know the differences between taxonomies and terms, don’t you? (see https://developer.wordpress.org/themes/basics/categories-tags-custom-taxonomies/)

So for example, if I registered custom “tag” and “category” taxonomies named gd_place_tags and gd_postcategory respectively, then I would display the tag/category list like so:

Tags: [bd_terms_list taxonomy="gd_place_tags"]

Categories: [bd_terms_list taxonomy="gd_postcategory"]

But if you’re absolutely certain the above just doesn’t work for you, then yes of course, you could just create another shortcode which calls the get_query_terms_list() function. 🙂

However, if I were you, I would try again with the modified bd_terms_list_custom_callback() function as well as the HTML code above, but I would double-check that I’m using the correct taxonomy slugs (and not the label like “My Taxonomy” or even the slug of a term in the taxonomy like my-category-in-my-taxonomy).