Yes, most people do start calling themselves a production company name without registering the name with a state or creating a corporation.
The reason why you should make a legitimate entity is for both tax reasons and other legal ones. You are liable for everything in the films, so an LLC (Limited Liability Corporation) is ideal for each major project. Investors from one film cannot pursue profits or losses from another project if they are each separate legal entities.
Kevin Smith has VIEW ASKEW, which is a parent company to several other LLC's, like TOO ASKEW, which was the corporation for MALLRATS. Now TOO ASKEW's sole deal was to sell the film to VIEW ASKEW, which in turn sold it to UNIVERSAL/GRAMERCY. This protected the individual people from having angry investors come after them personally for the financial losses when their next film was a hit.
When most people fake it and just use a production company name without actually forming the legal entity, they are DBA (Doing Business As), which essentially means it is an alias for the individual and they have no protection for their own personal assets for any losses or law suits, including copyright infringement, injuries, liability, etc.