Is it possible to have a template that works on multiple categories where the link address contains the specific category?

Why not using a meta field ? That way you can determine which template to use for specific category template.

In your loop just get the current category that you are trying to load:

if( have_posts() ) {
while( have_posts() ) {

    the_post();
    $meta_value = get_post_meta( get_the_ID() , 'meta-field', true );

    if($meta_value == 'template1') { 
         //I prefer using this method. Create folder templates with file content-template1.php where you can design your content. That way it is way more clear.
         get_template_part('templates/content','template1');
         // or you can use this method
         the_title();
    }

}

// Very Important
wp_reset_postdata();
}