It is very analogical scenario as connecting capacitors or electronic DC sources ( with floating potentials ) in a serie.
With 2 cells in a serie, the same provided charge spends the doubled amount of electrode material. And vice versa for charging.
Even without looking into electrochemistry details, energy conservation law requires doubled voltage for 2 identical cells in a serie.
In fact, such voltage doubling has nothing to do with electrochemistry.
Remember that electrolyte in cells has it electrostatic potential, and that thede potentials differ between cells by the cell voltage.
Potentials in cells, in the electrochemistry context, are floating potentials, related to potential of the standard hydrogen electrode, would be put into the cell.
In context of physics, the same electrode in cells in a serie, including the hydrogen one, has different electrostatic potential wrt (physical) infinity, what is conventional zero potential reference.
Electrochemical potentials of electrodes are electrostatic potentials related not to physical infinity as we are used to, but to (would be there ) the electrostatic potential of the standard hydrogen electrode in the same cell.
If cells are connected to a serie, electrostatic potentials of the same respective electrodes across the serie differ by the multiple of the cell voltage.