Not necessarily. Trademark owners are offered protection via a myriad of Rights Protection Mechanisms designed to prevent Abusive Registrations. Multiple legitimate trademarks can exist in many different jurisdictions for the same word.
You will want to check the policies related to each of our TLDs to make sure you are registering your name at the right time to maximize your chances of brand consistency. See: http://icmregistry.com/programs/launch-phases/