How does a color cycling LED work?
It could vary well be an MCU. It's incredible how small and cheap they can be manufactured these days.

But it doesn't necessarily have to be an MCU. There are plenty of ICs and hardware combinations of swtches that can accomplish such feats.

Imagine for example two switches with capacitive networks  combined in such a way that they charge and discharge each other over some interval, thus supply the two different LED color circuits.