I'd have to say that the onus does fall on the policyholder to ensure that his documentation shows the correct registration. The policyholder signs a declaration to state all is correct, and the insurer acts o nthis.
If the documentation you had is correct, but the broker still managed to submit the incorrect data to the insurer, then the fault lies with the broker, and it is they that are responsible - we (insurers) cannot be held responsible for being passed incorrect information - upmost good faith is the principle whereby we treat what the policyholder (or broker) tells us as gospel.
Obviously if the insurer subsequently updated MID incorrectly, then the onus falls on them. It happens - human error (I know it's not a good excuse, but it happens).
The insurer will inevitably pass you to the broker if the error came from them. As I say, if you gave the insurer incorrect info, would you expect them to cover the blame (well, some would). Whilst the broker is a representative of sorts, if they ******** it up, it's down to them to reimburse you.