Set term on an attachment using wp_set_object_terms and want to display the full term text but it’s showing a slug instead

When a visitor fills out the email address field with
“[email protected]”, what is displayed on the Attachment details screen
(as in siteurl.com/wp-admin/upload.php?item=xx ) for that email field
is “nameemail-com”.

Yes, and that is the default behavior, i.e. WordPress displays the term slug instead of name (or the actual email address in your case).

But there’s a filter hook you can use, namely attachment_fields_to_edit, if you want the term field to display the name instead of slug.

Working example:

add_filter( 'attachment_fields_to_edit', 'my_attachment_fields_to_edit' );
function my_attachment_fields_to_edit( $form_fields ) {
    $taxonomy = 'submitter_email';

    // Do nothing if the Submitter Email field is not in the fields list.
    if ( ! isset( $form_fields[ $taxonomy ] ) ) {
        return $form_fields;
    }

    // Get the term by its slug.
    $field = (array) $form_fields[ $taxonomy ];
    $term  = empty( $field['taxonomy'] ) ? null :
        get_term_by( 'slug', $field['value'], $taxonomy );

    // Use the term name.
    if ( $term instanceof WP_Term ) {
        $form_fields[ $taxonomy ]['value'] = $term->name;
    }

    return $form_fields;
}