It is simply up to the design of the battery to decide whether you can carry current by connecting opposite poles of two different batteries or not.
Usable batteries we see today are designed so the internal chemical potential builds up electric potential between terminals of the battery so they are opposite-charged. The conductive path (wire) between the opposite-charged terminals of the same battery drives the internal chemical reaction forward. When we contact opposite terminals of two isolated batteries this doesn't happen at a meaningful level (for power generation) because the external flow of charges through the wire is not balanced by the internal flow of charges.
There can be a still minor current flow as terminals and the
batteries can still have minor potential (because of the difference in potential between materials) between them
It is important to remember that we don't store all energy in the
battery as charges, but rather the flow of charges is driven by the
In a capacitor, we actually store all the energy as charges however we will not still see a meaningful current flowing between oppositely charged terminals of two isolated capacitors as internal charges are balanced out between the oppositely charged and closely structured (narrow gap between) plates.
However, if we were able to produce a hypothetical futuristic ultra-high-density (enough to call it a battery) electro-static charge storage device with a regulator and an electrostatic charge generator for charging, and if we actually made this battery (or capacitor) as a single device isolated-pole device (name made up by me), then we can generate a meaningful (to generate power) current to flow through a conductor connecting opposite charged poles of the two batteries or between the charged pole of the battery and neutral pole of the other battery(less practical). The physical feasibility of such an energy storage device probably would be unrealistic as the entire process (conductors and the load) and the battery would have to be enclosed in dense electrical insulation. And the design of the regulator and charge generator is to be considered too.
Note: Although we can make low-density charge storage devices, they will be like capacitors and hence will not be considered usable enough batteries.