I looked on the California Government website for information, but was unable to find anything about the publishing industry. (BTW, in case this makes any difference. I do not handle any purchases, as my books are sold on Amazon.)
I am not licensed in CA, so can only answer in general terms. While you probably do not need a special license, I cannot believe CA would allow you to operate a book publishing business withot a business permit. I doubt you can open a business bank acount without one. You will have to be in the state corporate income tax system. Will you have any employees? Then you need to be in the unemployment and worker compensation systems.
Here is a portal with general information about opening a business in CA.
You may also need to obtain a local business license.
It doesn't matter that you are selling through Amazon. When you do that, how do you handle sales ta? Does Amazon handle that for you?
Please see the guide linked below, or one like it, which is not unique to the publishing industry but has everything you need to strart a small business in CA.
You may also want to hire a business lawyer to make sure you've formed a separate business entity with appropriate tax status, registered an available domain name that is also viable as a trademark, liability insurance, etc. etc.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
I am not aware of any specific state license requirement for a book publisher in California. However, you will need a city business license in the are where the business is established. You may also want to look to see if there is a SCORE (Service Corps Of Retired Executicves)chapter in your area for assiastance negotiating the regulatory reqiuirements of your area.