How are bilge keels similar to the wings of an airplane?

They are NOT very similar, although they may have a streamlined aerofoil cross-section (to reduce drag).

Bilge keels (& most keels) serve to stabilise the vessels course and to prevent drift to leeward; so in some ways are similar to an aircrafts's fin (UK) or vertical stabilizer (US).

The secondary function of a bilge keel is to allow the boat to settle in a stable manner at low tide.