The story
Back-formation from mentor. Although mentor comes from Ancient Greek Μέντωρ (Méntōr), the name of a mythological figure, it was mistakenly analyzed as terminating in the suffix -or (“doer”), leading to a form using the French patientive suffix -ee on the model of pairs such as donor–donee and employer–employee. Attested since at least 1958.
Etymology from Wiktionary, CC BY-SA 4.0.