But there presumably must have been a time when you would have had to go to great lengths to explain to someone what an assumed identity was.

That time is clearly before the Arthurian cycle, which contains several instances of knights taking someone else's armour and being taken for that person - most famously, Kay the Seneschal and Lancelot. Arguably also before the period in which Greek myths were composed; Zeus occasionally disguises himself as someone's husband for purposes of seduction. In the Bible, Jacob disguises himself as his brother Esau to obtain their father's blessing, although admittedly the deception hinges on their father being blind. Mistaken identity seems to be a fairly old concept, then.