Electronic – What ESP8266 module is this

There are three reasons this can happen.

1. Fake IC. Either not actually a regulator, or it's specs do not match what you were told.

2. Damaged IC. Static electricity, Reverse Voltage, or some other reason. Not always visible damage.

3. Heat. Heat kills. Under-designed applications may not allow full use of a part's specs.

In this case, it seems to be reason 3. A Sot-23 IC with little to no copper pour or external heatsink to keep it within it's allowable junction temperature. If you don't take this into account, then it will fry itself. Some ICs may have over-temperature lockout circuitry, but don't count on it.

Here, the issue is highly susceptible to Voltage and Current concerns, re: Power/Wattage. As a Linear Regulator, the input voltage * input current is wasted in heat. So while it could do higher voltage at a low current, or a high current at a lower voltage, it can't do both.

The application, a ESP8266, will draw up to 400 mA at 3.3V. With a 12V input, that's (12V - 3.3V) * 0.4A = 3.48 Watts. Too much for no heatsinking.

This application really only allows for 5V input. (5V - 3.3V) * 0.4A = 0.68 Watts. Much more manageable.

As OP has shown, 3.48 Watts was enough to UNSOLDER the part from the board. Reverse reflow soldering ha. The first board had physical damage instead.

The Regulator itself would most likely actually be 12V compatible, with careful selection of input voltage and current requirements. If you hold the ESP8266 in reset, and power something small, like a 20mA LED, you will see it working just fine from 12V.

You are missing a GND connection between ESP and Arduino when you're powering the ESP externally. In general, devices which share voltage-based signals also have to share common ground. Otherwise, without a common reference voltage becomes meaningless: a signal of +5V on Arduino may be seen as -1V on ESP (assuming there's 6V difference between ground voltages of the two).

You can also reason in terms of current: when Arduino drives its TX pin high and a current starts flowing, how does that current return back to Arduino?