How do I get my suggest to only suggest one taxonomy?

After join term_id exist two times. You should specify which term_id you are querying by prefixing full table name (or alias).

Also there is a other error term_taxonomy.term_id = terms.term_id missing table prefix.

So the updated query will look like

$query = 'SELECT term_taxonomy_id, ' . $wpdb->prefix . 'terms.term_id, taxonomy, name FROM ' . $wpdb->term_taxonomy . '
    JOIN ' . $wpdb->terms . ' ON ' . $wpdb->prefix . 'term_taxonomy.term_id = ' . $wpdb->prefix . 'terms.term_id
    WHERE taxonomy = \'category\'
    AND name LIKE \'' . $search . '%\'
    ORDER BY name ASC';