The analysis is the same. California recognizes the concepts of "general" and "special" employers, as well as joint employers. In most cases, you will be an employee of the agency, as well as the client. But I agree with my colleague: talk with an employment lawyer about the specific facts of your case so he or she can determine which classification likely applies to you.
Craig T. Byrnes
Disclaimer: Please be aware that I am not offering legal advice, nor forming an attorney-client relationship with you. I am not representing you, nor doing anything to protect your legal rights. If you believe that you have suffered a legal wrong, take action before any statute or limitations expires, or your right to do so may be lost forever. Good luck in your legal matter.
You should consult an employment lawyer, who can directly answer your questions based on specific facts. Your question seems to conflating multiple discrete issues:
1) Whether a person should be considered an "employee" versus an independent contractor?
2) What rights do employees have under state and federal law?
3) Whether an employee should be considered an employee of the employment agency, the employment agency's clients, or both?
Here is a helpful article about dual employment and the "borrowed servant doctrine":
This answer is not a substitute for legal advice and it does not create an attorney-client relationship. Seek the advice of a licensed attorney before taking any action that may affect your rights