In the trailer for the new Assassin's Creed game, there is an interesting line at 1:27 where the narrator refers to your character "writing his odyssey across the mountains and the seas".
Of course "Ancient Greece" casts a huge net in terms of timeframe, but I believe that the game is set in the Classical era, since Socrates is one of the quest givers in the game. Would that have been enough time for the term "Odyssey" to become a more generic term, rather than just referring to Odysseus' journey?
It seems to have been a relatively recent shift, at least in English. The Oxford English Dictionary lists the first usage of 'Odyssey' in the sense of a long, adventurous journey as being in 1886 - in Kidnapped by Robert Louis Stevenson no less! Before then, in English, Latin and Ancient Greek, it seems to only ever be used to refer to the poem by Homer.