The external EVSE is a semi-smart power switch. All of the real smarts and the ACV to DCV conversion is done in the car. The EVSE announces to the vehicle how much current it has. It does not say how much voltage. The car, as long as you don't provide power outside of what it can handle, doesn't really care. The EVSE sends a signal that says - I can provide up to X amps. The car then dutifully never tries to draw more current than the source says it has. Amps*Volts=POWER. So, the higher the voltage OR amps will provide more power.
You can think of the car as a big power user that can adjust so that it doesn't pull more power than what is actually available. The EVSE tells it, and does some interlock safety stuff, but basically is just a smart power switch.
It's possible for an EVSE to sense the power available and potentially say it has more amps, but it would require an adapter based on the plug attached to it. In the USA, a plug with the standard 15A configuration will fit into a 15 or 20A receptacle, and it has no easy way to tell whether it's plugged into a 15A or 20A one. If it has the 20A plug, it assumes things were wired properly, and it's in a 20A receptacle. Doing this manually with a switch is dangerous, as you could have it set for a 20A circuit, and plug it into a 15A one (assuming it had the 15A plug), and you'd overload the source. So, there's no way for the EVSE to really know what it's plugged into.
Those that are 120/240 v compatible (that requires an internal multivoltage power supply), so that it can generate and process the right interlock signals, not to handle more current. There's more power on a 240vac device at the same amperage...remember, power=volts*amps, so if you double the voltage, you'll get double the power, but the amps stays the same. The only way the car will try to draw more amps, is if the EVSE announces it has more.
Some EVSEs can be internally adjusted to announce they have more, but you must be fastidious to ensure the wiring can handle how you're setting it.
2011 535i x-drive GT, 2014 i3 BEV
Soon (hopefully!) A 2021 X5 45e will replace the above