Wordpress - Get The Post Type A Taxonomy Is Attached To

If we peek into the global $wp_taxonomies variable we see the associated object types.

There might be better ways to do this or even core functions, but you could try the following:

function wpse_172645_get_post_types_by_taxonomy( $tax = 'category' )
{
    global $wp_taxonomies;
    return ( isset( $wp_taxonomies[$tax] ) ) ? $wp_taxonomies[$tax]->object_type : array();
}

then for the default setup you get:

$out = wpse_172645_get_post_types_by_taxonomy( 'category' );
print_r( $out );

with the output:

Array
(
    [0] => post
)

You can do the reverse with get_object_taxonomies. Combine it with get_post_types to iterate over post types to check the taxonomies registered for each.

EDIT- Here's an example that produces the same output as @birgire's function, without using dirty globals.

function wpse_172645_get_post_types_by_taxonomy( $tax = 'category' ){
    $out = array();
    $post_types = get_post_types();
    foreach( $post_types as $post_type ){
        $taxonomies = get_object_taxonomies( $post_type );
        if( in_array( $tax, $taxonomies ) ){
            $out[] = $post_type;
        }
    }
    return $out;
}

There is no need to use the $wp_taxonomies global variable directly as WordPress has the get_taxonomy() function to access taxonomies in it ( in core since version 2.3.0). Since WordPress 4.7.0 the return type is a WP_Taxonomy object with the property $object_type

$taxonomy = get_taxonomy( 'category' );
print_r( $taxonomy->object_type );