Compare: Primary risk of overcharge of a cell is catching fire.
Part of Cell degradation.
Overdischarge leads to crystal forming of Pb (or Lithium plating, if we are talking about Lithium-ion cells), which reduces capacity and increases impedance, while overcharge leads to disassociation of oxides with oxygen gas production.
Overdischarge can also create irreversible lithium oxides.
The typical anode material is carbon, and it's stable with or without the Lithium ions.
When a cell is severely discharged (to about 1.5V), at least locally, corrosion of copper current collector happens, releasing positive Cu+ ions into the electrolyte.