We already do.
We give licenses (and CA ID's) to legal aliens -- both residient and nonresident. We also honor foreign licenses. Past bills by Cedillo (SB60, AB60) were specifically for illegal aliens, and SB1160 is no different. Cedillo's bill has nothing to do with legal aliens.
It helps us keep track of them,
I don't know if they share address information with the USCIS (formerly called INS). Once you get a CA ID card, you are not required to update your address if you move, and it certainly won't update itself!
it encourages them to get insurance, all of which help us as a country.
Insurance and licensing are separate issues, since insurance is for the car but licensing is for the driver. Also, state ID cards have nothing to do with driving or insurance.
Some illegal aliens already drive without a license or insurance. Why would they buy insurance if they feel they don't need it now?