Recently, I bought a USB car charger that shows my batter voltage at about 13.7v when the car is running. Is it possible that this is accurate? Since you can get a lot of gimmicks on eBay, I want to be sure if this is even possible.
A car cigarette lighter or 12V power outlet in a car is, like most everything else, connected directly to the battery positive/alternator output, through a fuse (typically 10 Amp). Sometimes it is connected through the main switched power relay. In any case, aside from the negligible voltage drop across the fuse and wiring, is an accurate measuring point for the voltage in the car. As mentioned, 13.7 is a typical voltage for a car under load while the alternator is running.
As to the accuracy though, keep in mind that the trinket you got is a cheaply made mass produced item. It will have a resistor divider to bring the voltage down to the microcontroller voltage level. But the lower the quality of the item, the more likely that the resistor is not a very precise, and the Analog to Digital converter in the microcontroller they used will not be very accurate. For the most part, this may mean it's off by a fraction of a volt. Not precision laboratory grade. But good enough for your average user.
Sidenote, I use one in my car due to alternator issues and it's as accurate as my free harbor freight multimeter.
13.7V is in the normal range for the 12V system in a car.
It will vary over 12.5 to 14.5v when the car is running and down to about 11.5 when it isn't.
The cars cigarette lighter is connected directly to the cars 12v system but as others have mentioned may vary slightly from the battery voltage due to the resistance of the wiring.
Depending upon the car the lighter connection may be active when the ignition is switched off or not. (Japanese cars tend to have them switched by the ignition, American cars tend to have them always active). They are usually fused for 10-20A.
First of all the voltage at cigarette lighter jack is always somewhat less than actual voltage on battery terminals. There are ground losses thorough car's chassis, losses of "positive" as it goes through couple of relays, fuses and wiring. Any load (like stereo, GPS, dashboard illumination, etc.) drops a bit the voltage you will read on the cigarette lighter jack.
Second - you can't be sure of this meter's accuracy. I don't know the exact product, but most chinese meters may have difference of about +/-0.3V.
Third - you don't have to worry much. The main purpose of such a meter is to know does you alternator charge or not. If you see anything over 12V-12.5V it charges and in most cases it charges OK. If your alternator fails and stops charging you will read values of about 11-11.5V or less.
Of course there are cases where alternator can charge poorly and the actual battery voltage is 13.2-13.6V or can overcharge (more than 15V) but it happens quite rare. Although rare, I recently had both of these problems on my Ford Mondeo MK3 equipped with Visteon alternator. This kind of fault can be seen by measuring battery voltage directly on its terminals. This means you should make additional wiring just for the voltmeter.
Another solution for a modern car (over 2000's) is to buy an OBD display. It shows you some stuff read by digital bus from the Engine Control Unit (ECU). ECU knows battery voltage with much better accuracy compared to what you measure on the cig.lighter. On some cars ECU has digital comunication with alternator's charging regulator.