What happens if you live in DC and transfer the title of the car to your daughter who lives in California? How will DC know not to bill me for registration?

You will have to transfer the title and then the new owner will need to register the car and get tags. You might be able to add the person to the title rather than transfer and then you might be able to keep the same registration/tags. But then you will both be on the title. And if it is a gift to certain family members it will be exempt from the excise tax the new owner would normally have to pay. See this form.
https://dmv.dc.gov/publication/excise-tax-gift-exemption-form

How would the DC government know who holds title?

I'm a DC resident with a DC registered car and would like to give my car to a family member also at my address. Title I think is pretty straightforward. Can I just "transfer" registration, too? Keep the existing license plates, etc.? Thank you for your advice.