How can a battery have different currents but the same voltage?

It is actually the charge capacities, that makes the difference between your different sizes of batteries. Charge capacity is often rated in ampere-hours.

All batteries inevitably have an internal resistance to them. They will not produce their nameplate voltage (open circuit voltage) at all currents, because there is inevitably a voltage drop across the internal resistance that you cannot remove. The less internal resistance, the more current a battery can deliver, closer to its theoretical output current for a given resistor.

Bigger batteries have more area for the contact of the electrochemical reactions, and thus less internal resistance.

For a given resistance, the same voltage will produce the same current, IF the power source can supply it. A battery (actually a cell) has internal resistance and a capacity related to its size. As current is drawn, some of the voltage is dropped across the internal resistance so a lower voltage is available for the external circuit.

A circuit that can be powered by an AAA battery, for example, will operate for a much longer time if it is powered by a D battery instead.

A device that draws a current of 1 amp from a D battery, with a typical capacity of 8000 mAh, would theoretically last for 8 hours. With an AAA battery, with a capacity of 540 mAh, it would last about 32 minutes. In a practical circuit you would not use an AAA to provide a current of 1 amp.

In any battery there is an internal resistance(IR) associated with the chemistry.

the AAA has the highest IR then the AA then the A, B, C and D.

as the battery gets bigger the effect is like putting the IR's in parallel reducing the resistance and increasing the available amperage or power.

This is a highly simplified explanation.