Britain was not at war with Denmark until the League of Armed Neutrality of 1800-1801 when the Baltic nations banded together to oppose British naval policy. We were at war with Denamrk again from 1807-14. So your ancestor could not have been an official Danish prisoner of war during the early 1790s.
However, if he was a POW, then he may have been taken captive on board a French vessel, such as a merchant ship or privateer, in the early years of the Revolutionary War. If so, he would have been classified as 'French', i.e. taken in the service of the French. His record will be in one of the many General Entry Books for French prisoners for the local depots at Stapleton,Bristol (ADM103/411, 412 and 419),Pembroke (ADM103/103,104 - though these are mainly for the later years from 1797). He would only be on parole if he was an officer.
As a Dane, if he was taken prisoner serving on board a French ship, he could apply to the Admiralty for release as being a subject of a neutral nation, and then decided to settle in the area, hence marrying a local girl in 1796.
The other possibility could be that he was a seaman on board British merchant vessels, as such ships would employ any skilled seaman no matter what their nationality.
The upshot of my reply is that you will need to find positive evidence that he was a prisoner of war before narrowing down the search to the particular ADM record that would shed light on his story.
Hope this is of interest. Good luck with your research.