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When signing in to a Nintendo Switch using Joy Cons, the screen will always show the color for them. But, how does it know? In terms of parts, all Joy-Cons seem just about interchangeable?
More importantly, can a user potentially change what color a Joy Con thinks it is?
This is a known design flaw which is only effecting early models of the Switch.
It's something to do with electrical shielding (or something technical).
First try these steps to make sure you are using the Joy Con correctly
Joy Con Not Responding or Responding Incorrectly When Used Wirelessly
If that didn't work, talk to Nintendo and they will "fix your problem for free" - which means you send in your joy con and they fix it and send it back. This can take up to 2 weeks.
Here are some helpful links for support:
Contact Nintendo Support in America
I'm surprised nobody has done a teardown video of the strap accessory yet, but you can in fact disassemble it and remove the strap. I don't have pictures, but here's the process.
WARNING: Performing these steps may void your warranty. There are several tiny pieces that may be lost or damaged during this process. Also, the white locking tab may be a looser than normal without the strap in place.
Best Answer
When the Joy Con connects to the Nintendo Switch via Bluetooth, it sends some data as a "handshake" between the console and the controller. In this data, the Joy Con itself reports the HEX Color and Side (L/R) of the controller.
This can be confirmed with a program called Joy-Con Toolkit that allows you to see and change the colors reported by the individual controllers just by connecting them via Bluetooth: