No. There are several levels of trademark protection. A mark can be registered at the state level, or the national level with the USPTO. Protection abroad would require registering the mark in individual countries where the party might prospectively do commerce.
The answer to this comes down to the financial feasibility of registering the mark in the various individual countries. If you plan on doing commerce internationally, then registering a mark may benefit the brand.
One major difference is how trademark rights are established. The U.S. reserves rights to the first person to use the mark, even if that person isn’t the first to register it. This is true in Canada, Italy and others.
But China, Mexico, and Vietnam grant rights to the first to file. It regularly happens that US owners of brands go to China to seek registration, only to find someone else has filed for the brand.
Other countries, like India and the UK, will not allow offensive trademarks.
It’s an international treaty designed to simplify the international trademark registration process. Registrants can file a single application, which will be applied to 122 member countries. You can read more on the World Intellectual Property Organization page here.
The obvious advantages are the simplified application process. Submitting a document in your home language is also easier than trying to submit forms in multiple countries, some of which may use languages you don’t understand. There is also one application fee, but be aware registration fees will still need to be paid for each country where the mark is registered.
The disadvantages are that when an application meets roadblocks in the individual countries, local counsel may need to be hired to assist with responding and completing registration.
The list of participating countries can be found here.