> What are the chances that I could find the exact charger needed for a GameBoy Colour?

Fairly high. Nitpick time: The Gameboy Color[0] took a standard-size battery that you can still buy today. It did not need to be charged, but you did have to turn the system off to swap batteries unless you had a barrel-jack adapter.

Barrel-jack DC wasn't quite standard, but you might be able to find something compatible if you went to an electronics supply store and paid careful attention to the listed input voltage and polarity on the device. Regardless, most people didn't bother tethering their Gameboy and just fed it batteries since it ran forever on them.

The real proprietary hellhole started with the Gameboy Advance SP, and didn't end until the Switch used Type-C. Hell, the SP is basically a modern smartphone:

1. Proprietary form-fitting battery pack

2. Custom power input connector

3. No separate headphone output

Bonus points: the headphone adapter Nintendo sold for the SP didn't have a power pass-through, so you had to choose between headphones or charging. Though there are third-party ones now that do both headphone output and USB-C power input.

[0] No "u", not even in the UK

To add insult to the injury, the charging port of Nintendo DS Lite was almost mini-USB, but not quite - with enough force and some metal bending it was apparently possible to make it work with regular mini-USB cable. Absolutely idiotic design, especially since that port was just 5V input anyway, anyone with soldering iron could tweak USB-A cable to work with the proprietary charging connector.