No, you do not need a license to apply eyelash extensions in the state of California. Similarly, you can be a makeup artist without a cosmetology license. You would only need a business license. Freelance makeup artists and freelance lash stylists do not need a cosmetology license.
You do not need a drivers license unless you want to drive a car on public roads. In that case you need a license because all states have laws that require you to have a license. That license proves you have taken a drivers test and are qualified to drive.