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.
It is idiomatic English, but it has a decidedly different meaning from 'They are cuddling in bed.'
'cozy' is an adjective (unless used as part of the particle verb 'cosy up', which can be contextually synonymous with 'cuddle' but is usually not at least where I'm from). So, in 'They are cozy in bed', you're saying that the adjective 'cozy' describes their current state of existence. Rather interestingly, this doesn't even require that they're in the same bed (it can contextually imply it, but does not require it to be the case).
In contrast, 'are cuddling' is a verb in present continuous tense, so 'They are cuddling in bed' describes that something is happening. It's also technically expressing a state of existence, but in a rather different way. It also does not require that they are in the same bed, but it implies it much more heavily than 'They are cozy in bed' does.
A slightly more concrete and clear-cut example of this would be the pair of sentences 'He is high in a tree' and 'He is climbing a tree'. They both describe a person being in a tree, but the first comments on where he is while the second comments on what he's doing there.