Here's how it works in Florida. Your state may vary.
People get licensed.
Companies get qualified.
If you are Bob and you have a license, you can contract for construction work. You can also start a company and qualify that company. Whether you call it Bob's Electric or Martha's Apple Pie Company, as long as you qualify it with that EC license, it can engage in construction contracting. You can qualify Jim's Electric as long as you have constructive control of the operations and make sure Jim's Electric complies with all laws (code compliance, workmen's comp compliance, etc.). You are on the hook for everything Jim's company does. You have to be very careful qualifying someone else's company. You can get burned and it can cost you your licensing privileges for life.
Even though you qualify a company, whether it's your company or someone else's, you can still, as a natural person, engage in construction contracting.
What you can't do is qualify someone else's company and your own company at the same time, because each LICENSED PERSON can only qualify one COMPANY.
So each license can allow one PERSON to engage in construction contracting and allow one COMPANY to engage in construction contracting.