Trademarks only restricts the same competitors in the same industry from using your registered brand name, or company name. Multiple registration by the same or different companies, each registration in different class of business are allowed. In accordance with the above subject the answer is yes of course. You can very well Trademark the same word mark of company based in different country as Trademark is territorial and it is only country specific. But in some case, the other company with substantial market reputation tends to oppose the Trademark registration in different country where they don’t have presence as well.

A seasoned trademark attorney would be the best person to ask to regard the situation. That being said the USPTO, frequently rejects trademark based on “likelihood of confusion”, that a consumer may face between a prospective trademark and one that is already filed with the USPTO. According to the USPTO, “the likelihood of confusion, exists between a trademark when the mark are so similar, and goods and services for which they are used are so related that the consumer would mistakenly believe they come from the same source. Each application is decided on its own fact, and no strict mechanical tests exists for determining the likelihood of confusion.”

• Aside from being rejected for having a similar textual composition to an already existing trademark, a new registration may fail for also sounding alike, looking similar or creating “the same commercial impression in the consuming public’s mind.

• The USPTO looks at whether a likelihood of confusion would exists as to the source of goods and services, rather than actual goods and service are likely to be confused.

• The USPTO cannot perform pre-application searches, or provide a advisory opinion regarding registration.

• If a trademark is registered in some country it will be protected only in this country as trademarks are territorial. However, in some cases a company may oppose a similar trademark abroad, this depends on a country. Let’s say that in 80% of the cases you can register the same wordmark, provided that there are no identical or similar trademarks in your country.
Businesses often seek the assistance of the lawyer to make sure the application gets accepted and to avoid any future potential legal liabilities. Trademark offices do not look at whether a mark under consideration is already trademarked in another country. Trademark registration is on a per-country basis, there is no international scheme.

On Contrary

Depends on the country. Trademark are country-limited, this means that if a trademark is registered in the US, it won’t be protected in China. Usually, if someone in China uses a trademark identical to a US trademark, he will not commit a trademark infringement, provided it is not registered there. Even Trump lost his trademark in China and it took him almost 10 years to recover it (he managed to recover it after becoming President of the US – that’s what we call the hard way to get back a trademark). However, in some common-law countries there is the “first to use” rule, which means that the one who used the trademark first, owns it (the US, Australia, Canada, the UK and so on). In this case, if a company with a foreign trademark, uses it in one of the countries, it will have it protected there. In the countries, which adopted the “first to file” rule, if a trademark is not filed, it isn’t protected. If you plan to do your business abroad – register your trademark.