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
)