Little is known about Brünnich but it’s possible that he was the son of a well-known Danish zoologist by the same name. If so, that would mean his grandfather was also a portrait painter, so perhaps this inspired him.
The elder Morten Thrane Brünnich (1737-1827) sat for the great Jens Juel (1745-1802) who was one of the foremost portraitists of his generation. So you can imagine the young Morten being surrounded by a high standard of artists during his childhood.