When I grew up, in Liverpool, my Mum would do her daily "shopping" in the shops that lined either side of the main road. If she ran out of something unexpectedly and needed it there and then, rather than wait for the next time she went shopping, I would be sent to the corner shop "to go for a message".

As said above, "message"= "errand". The difference between that and "shopping" was a matter of scale.

DC