They don't have to be precisely equal. However if they are not, that means there is an overall charge. In electrical terms, that overall charge is on the self-capacitance of the capacitor. Let's put some numbers on it.
Take a 100uF capacitor. Place 1,000 microcoulombs on one plate and -1,000 on the other. The capacitor is then charged to 10v and there is no left-over charge. But now add a further charge of 1 nanocoulomb to one plate. That's one part in a million imbalance. A typical small capacitor might have a self-capacitance of 1 pF. The nanocoulomb raises its potential by 1000 volts - 100 times the plate-to-plate voltage.
You also have what are usually called "stray" capacitances, such as from the body of the capacitor to adjacent conductors, be they "earth/ground" or something else. These may very well be a bit larger than the self capacitance but if we say 10pF, that still leaves the one part per million imbalance raising the overall potential by 100 volts.
So there is no absolute law saying the charges are the same. However if they are not, there will be huge voltages trying to get the imbalance away. Besides which, you are most unlikely to want to operate a capacitor like that.
No, you can't just get rid of a charge "into the battery". Sometimes garage mechanics don't use scientific terms very accurately. To a mechanic, putting charge into a battery means sending a current through it. So the charge that goes in is equal to the charge that comes out. A battery doesn't just magically absorb charge.