"any type" of switch is not quite true. The coil draws a fair inrush current and, as an inductive load, has a high kickback voltage. You get some sparking both on connect and disconnect. The Honda starter switch is very small but is actually a pretty robust switch, I would guess about 10A capacity.
A switch rated 3A AC resistive load (typical common switch) might weld itself ON. This is pretty common. Or it might just burn out.
The difference between resistive and inductive load is the inductor's characteristic high inrush current and kickback voltage on disconnect.
To be safe I would get a push button switch rated 10A DC inductive load. This won't be cheap - or available at Radio Shack. Newark or Digikey will have them.
An aftermarket automotive ignition keyswitch is designed for exactly this application and, if it is an acceptable option for you, is probably the best choice.