This isn't entirely correct. Being an official Union with a negotiating license is really just for show. Ultimately a Union represents a number of people. If that number is large enough, or their importance large enough, then employers, and Labour Courts, listen to them.You cannot form another union as you would not get a negotiating licence...well technically i suppose you could but as no one would negotiate with you it wouldn't be much of a point.
At some point there were no Unions - then there were Unions. These weren't offical Unions with licences and seats at the negotiating table with Bertie and the boys. They were common people who stood up for their rights.