Two unrelated Latin words: 'pālus' (stake) for the stick, 'polus' from Greek 'polos' (axis) for Earth's endpoints.
A long thin piece of wood or metal; either end of the Earth's axis.
The 'stick' sense from Old English 'pāl' meaning 'stake, pole,' from Latin 'pālus' (stake). The geographical sense from Latin 'polus,' from Greek 'polos' (pivot, axis). Two different Latin words merged in English. Key roots: pālus (Latin: "stake"), polos (Greek: "pivot, axis (geographic sense)").
'Pole' (stick) and 'pole' (North/South) are completely different words from different languages — both Latin, but unrelated roots. 'Impale' is from the stake-pole.