Origins
Flavour was once about the nose, not the tongue. The word descends from Old French flaor meaning 'smell' or 'odour', from Vulgar Latin *flātor, from flāre — 'to blow'. A flavour was something borne on the air, blown from food to nostril.
The shift from smell to taste is not arbitrary. Aroma and taste are neurologically intertwined — most of what we call flavour is actually smell. When you have a cold and food tastes bland, your tongue still works; it is your nose that has shut down. The etymology anticipated the neuroscience by centuries.
Latin flāre produced an enormous family of English words, all connected to blowing. Inflate means 'to blow into'. Deflate means 'to blow away from'. Flatulence is 'an unwanted blowing'. Soufflé, from French souffler ('to blow'), is literally 'something puffed up'.
Figurative Development
Flair belongs here too, through French flairer ('to smell out, to sniff'). To have a flair for something is to have a nose for it — the ability to detect what others miss. The metaphor is olfactory: instinct as a kind of smelling.
The British flavour and American flavor diverged in spelling during the 18th century, but both trace back to the same Old French form. The word crossed the Channel as a scent and became a taste — one of the tidiest semantic shifts in English.