How to get a separate form_id for each of the taxonomy/term/%term exposed form?
Without patching the module you can't...but the View object is always available in the form state, so you can use that to check where you are. e.g.

function MYMODULE_form_alter(&amp;$form, &amp;$form_state, $form_id) {
  if ($form_id == 'views_exposed_form') {
    $view = $form_state['view'];
    if ($view-&gt;name == 'foo' &amp;&amp; $view-&gt;current_display == 'bar') {
      // ...
    }
  }
}