Are the numbers on a shifter standardised? My bike has several gears labelled one to six. Do these numbers relate specifically to a certain standard? In other words, would a bike set to say gear four on one bike, perform in the same way as another different bike set to the same gear?
No. 

The full gearing ratio of the bike (from your foot to the tire contact patch) is determined by four factors: the crank length, the chainring diameter, the cog diameter and the wheel diameter. Since these components can be changed independently, it is impossible to develop a uniform numbering system for a single component ("uniform" in this case means getting the same full gearing ratio from the same gear numbers selected for the chainring and for the cog).

I.e. even if you standardize the "front gear number to chainring diameter" and "rear gear number to cog diameter" relationships, two bikes with different wheel diameters will still perform significantly differently even when the same gear numbers are selected on both of them.