It is possible to protect a name that is in use by a company offering service or goods that are different from your company. However, it is important to take customer confusion into consideration because if customers are likely to confuse your services or goods then you may not be able to register the same name. For example, in a recent case company selling furnishings was refused registration because a furniture company had the identical trademark.
Whether a trade name or trademark is protectable at all, or whether one infringes another, are very, very fact specific questions. No one can provide you with any actionable information in the abstract. That being said, if a graphic design company is doing business in New York as Oofy Goofy, Inc. then it's very likely an infringement of that trade name if another company using that identical name opens a public relations business in the state. Only your own New York-licensed trademark attorney can provide you with actionable information.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
Probably not, but I would need many more details before advising you. The issue is whether there would be a likelihood of consumer confusion. Because the two companies are in the same industry, it would seem that this could lead to consumer confusion. There may be exceptions to this however. Depending on what you mean by "different specialization" it may be that the two companies do not compete with each other for the same customers, or provide the same services, in which case the likelihood of consumer confusion may be reduced. One thing is certain---before you go down this road you need to retain and consult with IP counsel. Developing a branding strategy is one of the single most critical parts of building any business---and most small business owners who are savvy understand that investing in intellectual property counsel is critical to their success branding and marketing. Failure to have an IP lawyer on retainer is one of the biggest mistakes many small business persona make---and trust me about this---when clients come to me after being sued because they did not not invest in IP counsel it will cost them a whole lot more to get out of trouble than if they had retained me to prevent trouble long before.
I am not sure they are the same industry. But they keystone is likely confusion. If a customer, not an expert in, say, advertizing or marketing, might confuse the two then the answer is No, cannot register. And senior in commerce wins.
But the totality of facts is what is important. Talk with a TM attorney and lay out all the facts in confidence, not just what you think is important. Spend a few bucks. Protect your good name. Or find a good name.
Licensed in Maryland with offices in Maryland and Oregon. Information here is general, does not create a lawyer-client relationship, and is not a substitute for consulting with an experienced attorney on the specifics of your situation.
Just an additional thought to the other excellent answers here, a lot will depend on how strong the mark is. Very strong or famous marks are granted broader protection than weak marks, meaning the stronger the mark is, the less closely related the goods or services have to be in order to find a likelihood of confusion.