If your business name, phrase, slogan, tagline, product name is being used to identify goods or services with your business, then it is a trademark.
The registration of a trademark means that the trademark has been searched by your lawyers and the United States Patent and Trademark Office. There is due diligence performed. The goods and services are defined properly. Your trademark has been offered for public review and passed that review.
After that process, you can “trademark your name.”
The process, however, is important to understand what other marks exist in the marketplace for your goods/services. So, while you want to “trademark your name,” your lawyers’ job is to avoid liability while filing that application for a trademark registration.
Yes, you can “trademark your name.” Just remember it may already be in use as a trademark, which creates liability when that mark is not searched.
Here is a video blog on the the importance of performing a trademark search: