LED bulbs: can they function on both 12V and 120V?

What they said, except for the "never plug an LED into 120V."

I mean, you CAN plug an LED directly into 120V. Nothing stopping you. It'll be exciting for a brief moment. Then smelly.

Look at the specifications on the package for voltage requirements. Some LEDs don't even like 12V; they sometimes need much less.

As for current ratings, it may help to understand that the 12V system (and indeed household 120V power) is basically a constant voltage system. The voltage is held at 12V (nominally) and the current that flows through any device is determined basically by ohm's law and its impedance. With most electronic circuits, it's not quite a pure application of ohm's law as the current can vary based on lots of things and generally is not a linear function of voltage, but the general model applies.

Thus, you can wire any sort of 12V device up to the lighter plug, up to the current rating of the plug and the circuit it's on, and it will only consume whatever current it consumes.

The lights would be designed for a specific voltage or voltage range. It's theoretically possible, but very unlikely, that this range could include both 12VDC and 120VAC. I also suspect that there was some sort of plug-in 12V power supply in the store display.

Incidentally, there are such things as constant current circuits, where the current is maintained at a fixed amount and the voltage across the circuit is varied to maintain this current. These were sometimes used in the early days of electrical lighting particularly for arc lighting. In a constant current circuit, all the loads are wired in series, rather than parallel, and to turn off a load you shut it with a switch wired across its leads, introducing basically a short circuit, rather than opening the circuit with a switch in series with the load. (Current-mode signaling is also sometimes used in some controls and digital interfaces, often with a 20 mA loop current. Teletype machines, for one example, commonly were interfaced with a 20 mA current loop serial interface.)

Never connect low voltage LED lighting directly to 120V AC household power......they operate on 12V DC.

120V AC LEDs have a built in AC/DC transformer -----typically 30V DC

Light Emitting Diodes operate at very small voltages and draw little current as you have found out. For RV use, they are configured to operate at 12V but will endure a few volts more.

I'm going to guess that the store you visited had a display that was powered with a 12V power supply. The power supply was probably powered by the standard 120V AC mains power.

Yes, your cigarette socket will offer the 12V necessary to power an LED array (as that is usually how they are built, not a single diode).

BTW, 120V and 12V are not measurements of current. They are measurements of voltage. Current is typically measured in Amps (A) or Watts (W).