There’s a fundamental difference in sense between the two sentence constructions you presented. The second, “Have you received my message?”, is asking the target recipient to just verify receipt of the message, while the first, “Did you get my message?”, can be understood in two levels or ways. The first level is in the same surface sense as that of the first question—verifying the physical receipt of the message; the second level is the deeper sense—finding out if the target receiver understood the message. In oral communication, how the question “Did you get my message?” will be understood will largely depend on how it is enunciated by the sender of the message. In written communication, on the other hand, the ambiguity might need to be clarified between the sender and receiver of the message.
