The module's javascript doesn't appear to attach correctly to fields for taxonomies. This appears to be because of an odd case in the condition controlling when the javascript is attached to the form.
Needs work
3.0
Code
Not all content is available!
It's likely this issue predates Contrib.social: some issue and comment data are missing.
No it's not possible.
You could copy from the unique_content_field_validation module, which already supports Taxonomy (you can see the unique checkbox when creating a Vocabulary, so that you don't have to override base fields, which by default are not shown on the UI), but is not using AJAX.