OK, I just tried to replicate this setup, wired up my TCS34725 using a breadboard, exactly as you did in the image. When I plugged in my Uno, both before and after uploading the sketch, the LED on the sensor did not light up, the serial monitor still output the "No TCS34725 found" message, and after a minute, the chip was hot to the touch.
I'm not sure whether it was fried due to incorrect wiring or just was defective, but I'm starting to feel that one way or another, this is not a functional chip. I'll have to try soldering up another one and seeing if it works any better.