When a battery is connected to a capacitor, why do the two plates acquire charges of the same magnitude? Will this be true if the two conductors are different sizes or shapes?
Because, as part of an electric circuit, every electron that leaves one plate is matched by one entering the other plate. It is a consequence of Kirchoff's Current Law (KCL):
"At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node"
Ironically, KCL does not apply to the capacitor itself, because electrons enter one plate and stay there, while electrons leave the other plate with none entering. However, KCL does apply to every other point in the circuit, which ensures that the resulting charges on the plates are equal and opposite. The size and shape of the conductors has no effect unless they are very close to each other and/or very long, which would act as an additional capacitor in the circuit.
KCL is a special case of the principle of conservation of charge.
Think of Voltage as being like pressure.
If you imagine something elastic that can store some amount of fluid eg. a balloon, it will fill until the pressure in in it matches the pressure of the supply.
The size or resistance of the connections between the battery and capacitor do not stop that happening, but they will affect how quickly it happens. A high resistance will cause the capacitor to charge slowly, where a low resistance will allow it to charge quickly.
Think of the balloon filling through a tiny tube or a hosepipe - the pressure will still equalise eventually, but the time varies.