Taxonomy fields store term IDs, not the name of the term. In addition to this they store arrays of values.
so something like this /projects/?designer=new_studio
would need to look something like this /projects/?designer=22
(id value is just an example)
And then the query would need to look something like this
$meta_query[] = array(
'key' => 'project_designer_obj',
'value' => '"'.$_GET['designer'].'"', // quotes around id important
'compare' => 'LIKE',
);