Electronic – Christmas card with tiny screen

Those 4 wires do not go to the LCD, but to the touchscreen. It is most likely a 4-wire resistive touchscreen.

As the comments have suggested, it is unlikely that you can control the LCD glass directly with your Arduino. That said: you CAN put this to a new use if you put the effort required into the project.

You have already started. You need to identify the microcontroller and follow the traces from the LCD display back to the micro. Then use the datasheet to work out how the controller is talking to the display.

Then you need to identify the LCD segments. This will most likely require that you write custom code for the micro to run through all of the multiplex possibilities and note which segment lights up as you go through each one.

Finally, write new custom code that makes the microcontroller drive the display the way you want. For example, you could implement a serial protocol that listens for commands from your Arduino and displays the result of those commands on the display.

This is actually a fabulous learning exercise should you decide to do it. You will learn much from the process.