When one says, "I took my hat and my leave", that can be identified as a "rhetorical device" and labelled a "zeugma". Unusual uses of language draw attention, create emphasis, and what's unusual here is the use of the one word in a literal and a metaphorical sense.
I suspect the same thing is going on with "har hoste og vondt". I think one "har" en hoste in a different way from how one "har vondt".
Anyway, so one thing that the sentence teaches is that zeugmas work in Norwegian.
Another way to say this is" Jeg hoster og har sår tunge" The doctor is asking what your condition is, and you can answer with that. Legen spør: Nå, min venn, hva feiler det deg i dag? ( Now my friend, what is " wrong" with you today) and you answer: Jeg hoster og har sår tunge ( I am caughing and my tongue is sore.)