I'm seeing conflicting information on the etymology of aubergine. It's either a diminutive of auberge, meaning a type of peach, or it was borrowed into French from the Catalan alberginera, which borrowed it from the Arabic al-badinjan, which borrowed it from the Persian badin-gan, which borrowed it from the Sanskrit vatigagama.
The other auberge, meaning 'inn', comes from the Frankish heriberga, which means 'army shelter'. It's not connected to the other auberge—they're just homonyms.