I wouldn't say it's wrong, but it sounds odd. I think the sentence is talking about two people in a relationship cuddling in their own bed - something quite ordinary. To say "they are cuddling in the bed" puts the emphasis on the bed, so it sounds like they've hopped into a display bed in a shop or something.
Swedes please correct me, but I think "i sängen" is the natural way to say this in Swedish, and "in bed" is the natural way to say it in English, it just happens that one uses the definite form and the other doesn't.
Having said that, I think "in the bed" is a technically valid translation, and probably should be accepted, so it's worth reporting it.
I think saying sängen refers to it being theirs. From what ive seen before adding a the to the is very common. I.e people say : han tar av sin hatten. It just indicates its his hat
So does this mean to cuddle or to get cozy (which I take to mean comfortable)?