Yes, the basic meaning of the word "company" is simply a group of persons--as little as two.
"Please keep me company," please stay with me while I do something. The musical "Company" by Stephen Sondheim asks the question of why people get married--is it just "for company?"
In the old days, small firms might start out with a name like "J. Loxton Grinstead, Grocer." Then Grinstead's sons might join and it would be "J. Loxton Grinstead & Sons." Then they would hire other people and it would become "J. Loxton Grinstead & Company," or "The Grinstead Company." One could imagine moving on to "Grinstead Ltd." or "GIPSYS (Grinco International Provision Systems Ltd.)"
In the theatre, the "company" is the full set of people that put on a play, including the "cast" of performers and the technicians, business managers, etc.
In business, a "company" simply means a group of people operating as a business. Most other terms have specific legal and technical meanings that vary from country to country. In the United States a "corporation" has a specific legal structure.
The official name of Boeing, the aircraft manufacturer, is still "The Boeing Company," preserving what is now a somewhat old-fashioned name.