In theory only, two identical batteries have the same voltage (same tank height in your example).
Practically speaking, when we connect two batteries rated at the same voltage, a current (small or not) flows between them till their internal voltages become exactly the same (like in water case).
Also every battery has an internal resistance. In theory, identical batteries should have equal internal resistances. In reality, they don't have. Therefore, when two similar batteries supply a current to an external load, the current of the battery that has the lower internal resistance is higher than of the other one. In other words, they don't discharge (or be charged if they are rechargeable) at the same rate.