I feel like I'm missing something obvious. I know different plug styles are used based on a circuit's amperage (in the US) but is the same style of plug used for both 110V and 220V outlets of the same amperage?
No, the plugs are different for different voltages. There are many different NEMA receptacles for various combinations of voltage and current, and also a "twist-lock" receptacles. Some common ones are:
An easy rule of thumb to remember is that a NEMA plug/outlet with both blades parallel is for 15A service ... if they are vertically oriented (as in normal residential), it is 120V service, if they are horizontal (in-line), it is 240V service.
20A service has the blades at right angles to one another ... 120V has the neutral blade horizontal and the hot blade vertical, while 240V has it the other way around (although there is no hot and neutral for 240V service ... both blades are hot).
Unfortunately, Aaron's other response is incorrect ... there are so-called "universal" duplex outlets that accommodate both 120/240V 15/20A configurations. Leviton has their 5825W, and there is one made by a niche supplier that is slightly different in configuration: http://www.ehydroponics.com/cap-universal-x-plug-120-240-outlet-receptacle.html
Hope this helps!