Can two different planes share the same flight number?

I think you are confusing aircraft tail number with flight number - the former is a unique registration number for each plane, while the latter simply describes a unique route operated. When airlines operate flights on a codeshare basis, one physical aircraft could be flying the same physical route sold under different flight numbers too!

Some flights don't even use the same plane type every day. Sometimes flight numbers are used for different destinations on different days. Here's a current example on Flightaware.com: United 831. For four recent days, the flight operated from Washington Dulles to Cancun. Then for the following three days, the flight operated from New Orleans to Denver. That pattern repeats. On most days, the flight is on an Airbus A320 (for both of the unique itineraries), but on January 19, the equipment is listed as a Boeing 757-200.

Yes, the same flight number can be used to refer to multiple flights active at the same time. This happens fairly regularly if a flight is heavily delayed. The previous days flight may still be active the same time as the current days flight takes off.

"Direct" Flights

It is also fairly common that the same flight number is used to refer to more than one route - this is fairly common in the US.

When Continental first launched service to Edinburgh, the flight operated under the flight code of CO37, which operated Edinburgh-Newark. There was also a second flight, Newark to Orlando, which also used the flight number CO37. This was operated as a "direct" flight, and both destinations (Newark and Orlando) were advertised as being served from Edinburgh.

In some circumstances, if the Edinburgh flight was delayed by a few hours, then it could be landing into Newark as the Orlando flight departs.

Callsigns

This doesn't pose a problem, as flight numbers are only used by travel agencies and passengers.

In aviation, the airlines and air traffic control system instead rely on callsigns. These callsigns are often the same* as the flight number - Continental used the callsign Continental 37 for the Edinburgh flight. However, callsigns can be changed at very short notice, so if the Edinburgh flight was indeed running late, the Orlando flight would operate under a different flight code to avoid any confusion. This may be as simple as adding "B" to the callsign, to become Continental 37B.

*Conventions for selecting callsigns are airline dependent. British Airways and Easyjet, for example, use callsigns that have no relation to the flight number, while other airlines tend to mirror the flight number in the call sign.