Yes, it is possible to own a trademark in multiple countries, and many companies do. However, keep in mind that this will be most likely achievable only by owning separate trademarks in each of the countries since trademarks generally don't apply across borders (the only exceptions are multi-country jurisdictions, such as the EU or Benelux).
It is also important to remember that different rules apply in different countries, and where one country's IP office might accept your application, others might refuse it or require you to make changes affecting the level of protection you'll then receive in that market. In general, you also won't be able to apply outside of your country of residence without the help of a legal representative (but there are exceptions to this as well).
Last but not least, please note that when applying for a trademark in multiple countries, the cost of registration will also add up.