How to wire many bi-color LEDs to a single power supply, and switch them on/off individually

Since this is a constant-current sink IC, no resistors are needed in series with the LEDs.

The current is programmed using a reference resistor connected to one of the pins that sets the reference current used to drive the LEDs, and it is the driver chip's responsibility to make sure the correct voltage and current is applied to the LED. In practice, I have found this works well.

You could add a resistor in series, but this defeats the purpose of using this chip. It will increase your part count, increase your minimum VDD requirement, and if you need to increase VDD, then you will waste additional power.

Below is the figure used to determine your reference current for each resistor from the datasheet:

Now the RGB LED has drop voltage 3.2V on green/blue and 2.0V for the red color at 20mA. This would result in noticeable brightness difference of the LEDs

Why would it? A constant 20mA through the segment is a constant 20mA, regardless of the led Vf or Vs. The internal driver might dissipate more energy but thats mostly negligible.

If you already tested it and thing there is a higher current through the red leds, a silicon diode or two will bring the total Vf of the segment closer to the 3.2V, better than a resistor would, in a constant current driver setup.