Do you actually mean C-7 as that seems the most applicable?
California Code of Regulations
Division 8, Title 16, Article 3. Classification
C-7 – Low Voltage Systems Contractor
A communication and low voltage contractor installs, services and maintains all types of communication and low voltage systems which are energy limited and do not exceed 91 volts. These systems include, but are not limited to telephone systems, sound systems, cable television systems, closed-circuit video systems, satellite dish antennas, instrumentation and temperature controls, and low voltage landscape lighting. Low voltage fire alarm systems are specifically not included in this section.
The specific requirements and applications vary greatly from state to state but you first have to understand that there are different types and levels of licenses and your comment regarding structural modifications and permanent wiring relate to totally different licenses. The General Contractor has to have a G.C. license, the Electrician an Electrical Contractor's license and so on. Some states lump everything under only a few licenses, others do as California appears to and have numerous specialty contractor licenses as well.
I am not a lawyer, so I would definitely get a more qualified legal opinion on this, but based on what I see in the California Business and Professions Code it appears that it is illegal to advertise (which includes business cards) or offer contracting services for construction or renovation for any area of work for which there is a contractor's license classification unless you are either licensed in that classification or specifically identify on all such advertisements and offers that you are not licensed. There is also the following:
7028. (a) It is a misdemeanor for any person to engage in the
business or act in the capacity of a contractor within this state
without having a license therefor, unless the person is particularly exempted from the provisions of this chapter.
Given the above description for a low voltage contractor, that classification would certainly seem to apply to audio and video systems. I would definitely check into this. It would probably not hurt to be licensed in any case and you might want to see if it is a requirement, people may have just been assuming you had a license.
FWIW, it appears that in California the license is for the company although the test is taken by an individual and it looks like what happens if that individual leaves varies depending on the situation. But again, just my layman's interpretation.