They are not addressable. The limited color changing available is already hardcoded into the IC.

But there are very similar fairy lights that are addressable. I got a few, hacked off the controller and put it on a ESP8266 running WLED.

https://www.amazon.com/gp/product/B08FMJ81HH/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1

Trying my best here - but you've only got two wires. I don't think those are addressable.

Is it these?

The controller you have in your computer has the data pins soldered, that's the D+/- pins. If you're looking for a complete controlled system, you'll want WS2811 or WS2812 lights, like these. (You don't need waterproof ones, those are just a quick lookup - unless you plan on putting them outside).

Another way to check would be to see what the output dc voltage/amperage rating on that usb plug is. If it's only 1A, they likely aren't adressable.

The board being used in your 'goal setup' appears to be a ESP8266 or ESP32 development board.

Can you link the guide you're following?