We identify a strong structural similarity between the Gavagai problem in language acquisition and the problem of imitation learning of multiple context-dependent sensorimotor skills from human teachers. In both cases, a learner has to resolve concurrently multiple types of ambiguities while learning how to act in response to particular contexts through the observation of a teacher's demonstrations. We argue that computational models of language acquisition and models of motor skill learning by demonstration have so far only considered distinct subsets of these types of ambiguities, leading to the use of distinct families of techniques across two loosely connected research domains. We present a computational model, mixing concepts and techniques from these two domains, involving a simulated robot learner interacting with a human teacher. Proof-of-concept experiments show that: 1) it is possible to consider simultaneously a larger set of ambiguities than considered so far in either domain; 2) this allows us to model important aspects of language acquisition and motor learning within a single process that does not initially separate what is "linguistic" from what is "non-linguistic". Rather, the model shows that a general form of imitation learning can allow a learner to discover channels of communication used by an ambiguous teacher, thus addressing a form of abstract Gavagai problem (ambiguity about which observed behavior is "linguistic", and in that case which modality is communicative). Keywords: language acquisition, sensorimotor learning, imitation learning, motor Gavagai problem, discovering linguistic channels, robot learning by demonstration.