"I do not have a shoe on."
Translation:Chan eil bròg orm.
Agam is possessive, and in this context the 'I don't have' is more stating that something isn't there rather than you possesing something. Were that something there it would be 'on' you. On is the operative word, it's why there's the prepositional pronoun orm = air (on) + mi (I/me).
As for why Duolingo says what it says, I don't know, because it's in beta maybe?
Hope this helps