"Will this law prohibit a bicyclist from passing a car closely?

No."

https://bikeeastbay.org/GiveMeThree

This seems arbitrary and dangerous. A car being passed may not notice the passing cyclist in his mirrors, and attempt a lane change. Forcing the 3 foot buffer on the cyclist as well gives a chance for both cyclist and car to notice and avoid the collision.

Cars not noticing cyclists in their mirrors before they change lanes is the fault of the driver, not the cyclist.

Motorists should always check their mirrors before changing lanes. There could also be other cars there.

The onus is on the two ton shielded metal vehicle, not the bicycle.

Obviously if the driver hits the cyclist, the driver is legally at fault. I am suggesting that the law anticipate the inevitable carelessness of drivers, and build in at least a small safeguard. I see no good reason not to make the cylists abide by a safe distance passing rule.

You check to see which one was doing the passing.