Yesterday I purchased a NETGEAR WNDR4300 v1 for $15 Canadian to use with openwrt. They failed to tell me it had probably a German adapter and it had a us 2 pin outlet to 2 rod little thing on the end. The unit seems to work fine and I am using it to write this. I chose to use a 12V 2A dc adapter I otherwise had because I don't trust this set up, but they likely had used it that way for some time. The euro brick said 220-240 v and did not mention 120V. When I measured it with a DVM I got to my surprise 12.xx volts. This leads me to believe it was in fact auto switching.
How does this detection work? And can there be any issues with it? Maybe you are better off if you are the lower power standard?
Those things will automatically adapt to a wide range of input voltages, particularly at low load, but running them out of specification permanently or at high load is not a good idea. At half the input voltage, current draw increases two times to deliver same output power and you never know if the circuit was designed to cope with it.
There's no auto switching. Either it's a wide-range flyback (likely at that operating power) which means the transformer is wound to typically operate at normal loads down to about 90V ac, or it's got an active PFC stage that boosts the AC input voltage up to a DC bus voltage of around 380-420V. Active PFC is rare at low power levels, but that then allows the transformer stage to be made with much tighter design margins, which can be beneficial for efficiency, packaging and cost. It's generally a legal requirement (effectively, due to 3rd harmonic regulations) at around 65-75W output power.
One thing you might encounter is the primary-side cap is smaller for a 240V only power supply. This means its output voltage might have a nice 120Hz component on it, as the converter struggles to keep up with loads near the bottom end of the AC cycle. For some devices, there will be no obvious effect, as they rarely use the 12V dc output directly - it usually gets post-converted to 5V, 3.3V, 1.8V, etc. So an input of 12V dc with dips down to 8-9V may not cause an obvious effect, but the long-term effect will be that ripple current cooks the capacitors in the adapter and, if present, the ones in the router, as they end up carrying the full load current of the device at 120Hz.
90-265V is a common universal voltage range.