At a guess, the EEPROM may hold data used to adjust the brightness of the LEDs at some point in a display pattern. That way the micro can simply increment a counter, read a couple of bytes from the EEPROM at an address that corresponds with that counter value, and apply those bytes to various timers to turn the LEDs on/off to produce the required PWM, patterns etc.
I cant immediately think of any other reason to include something like that in something so cheap.
That also means you can use any manner of bargain basement microcontroller with minimal features, even if you have to bit bang some SPI or I2C to read the EEPROM.
If you have the tools, start with a dump of the EEPROM and see what it contains.