Welcome to the forums!
I think your project should be fun and I did a cheesy sketch of something that might be a starting point.
One problem is the aquarium LED controller board runs off 5V and the ESP8266 is a 3.3V MCU. So you need to not mix and match signals without using a level-translator.
You can see the Touch sensor IC has two open-drain outputs, which means you could add your two ESP8266 output connections to the aquarium LED controller MCU by using a pair of extra N-ch mosfets such as BSS138. So both could get along (ESP8266 and touch buttons) and control things.
I would also add a RESET output so you can reset the LED MCU because if the aquarium MCU gets out of sync, i.e. someone touches a button then your code does not know this happened - unless you also want to add hardware to read the touch buttons to follow.
Possible I/O assignment:
ESP8266_GPIO4 to assert button 1
ESP8266_GPIO5 to assert button 2
ESP8266_GPIO14 to assert MCU reset
Outputting a logic 1 (pulse) would turn on the mosfet which pulls down (to 0V) the STM8 MCU signal to assert it. All three are active low, meaning a logic 0 activates that line on the STM8. This is just from looking at the PC board and guessing how they did it.
I would confirm the LED power supply is properly isolated from AC mains and has extra capacity if you want to use it to also power the ESP8266. I guess it is maybe 9V or 12V which is too much so you need something to lower it to power the 3.3V ESP8266. If the aquarium LED/MCU power supply is properly isolated from mains (it should be) then you can tie the two circuit GND's together.
It's just that some really cheap aquarium LED setups use a "mains capacitive dropper" power supply which is not safe to connect to. It depends on how many LED's there are.