Is it normal for the controller box on permanent outdoor lights to get hot enough to melt?

That tan spot is not normal operating heat. The controller boxes do run warm, I can feel warmth on mine any night they're on, but warm plastic and plastic that's changed color are two different things. Discoloration like that means something got hot enought to cook the housing from the inside, and nine times out of ten that's a failed component on the board or a bad solder joint arcing. The fact that the lights still cycle colors fine doesn't tell you much, because the part that's cooking is usally on the power side, and the lights will keep working right up until they don't. I had almost the identical situation with mine, second season up, same faint burnt smell, same yellowed patch on the back of the housing. Lights worked perfectly. I figured it was fine and left it, and about three weeks later the whole string went dark mid-evening. When I pulled the controller down the inside of the case was brown around the transformer area. So in my experiance that early discoloration is the warning light, not paranoia. The wiring connections are worth a look too while you're up there. If a splice or a plug connection got loose or water worked its way in, that can cause heat buildup at the controller even though nothing else on the circuit complains. Shaded under the eaves is actually a spot where moisture can sit longer than you'd think. I'd replace the controller before the holidays rather than trust it through January. Since yours are only two seasons in, contact the manufacturer's support first with a description of the discoloration and the smell. Mine sent a replacement controller without much back and forth, and honestly it's cheaper for them to ship one than argue about it. Don't keep running it overnight or while you're away in the meantime, that's the piece I'd genuinely worry about.

I agree that the discoloration is a sign of a power delivery issue, not just normal heat. Whenever I've tested a setup like this for clients, the first thing I check is the voltage drop at the point of connection. If you're running a long string of lights, you can easily see a 5% drop across the run before the transformer, wich causes internal heating that looks exactly like that burn. I had a job where a loose ground connection caused the transformer to operate at 115 volts instead of 120. Even if the lights still work, that constant strain will cook the internal circuitry over time. Check your power input.

I definately agree with the point that the discoloration is a failure signal, not just normal operating heat. I had an exact match of this situation happen to me a couple of years ago. It wasn't a voltage drop issue, though. The internal wiring had just developed a loose connection where the circuit board met the terminal block. It was literally corroded and heating up right there. I ended up getting a soldering kit and fixing that specific bad joint instead of replacing the entire unit. It’s always worth checking those connections if you decide to look inside.