Control LED states (on/off/brightness) through multiple clicks of push button
The chip is either a specially made IC that does all of the functions in hardware, or it is a small microprocessor programmed to carry the the described functions.
Your options are to build similar hardware out of standard logic ICs, or to program a small microcontroller to carry out the functions you want.
Either option will be bulkier and more complicated than what you are looking at.
If you build it out of standard logic ICs, it will take at least a couple of ICs and some other parts.
If you go the microcontroller route, you'll probably have to add in a couple of transistors to help the controller deliver enough current to the LEDs.

If you go the hardware way, you'll learn more about logic and electronics and zero about microcontrollers and programming.
If you go the microcontroller route, you'll learn a bit about logic, a bit about programming, and a bit about electronics.