Yes, if you set up a business you can choose a business name.
Make sure the name doesn't look like the name of another business too much, especially if the other company is offering the same services or sells the same products as you will do. Your business name should also not look like an existing trademark. Your businessname should also not be misleading. So don't mention "and associates" if you don't have an associate.
If you use a name which already exists you can be confronted with claims from the company who has been using this name. This may lead to a court case in which is decided that you can't use the name anymore. There may not be an issue if you are operating your business in a different region or if you offer different services. Also if several different companies use the same name or use part of the name chances are lower that there will be a problem. But of course a unique name is preferred.
Expatax can of course also check the trade register for you, as part of the package we offer regarding setting up a limited company or other type of business. For more info check the brochures category.