Well, the last danish monarch to marry someone who wasn't a german noblewoman as Niels of Denmark in the 1130s, when he married Ulvhild Hakansdotter (who also married two different Swedish kings, Inge II and Sverker I). In Sweden, the last one was Eric XI of Sweden (but while his wife was the daughter of a nobleman, her mother was a Swedish Princess).
Morganatic and equal marriages are confusing (and often times don't make sense to me), but on this case (BTW, who are the parents of this probably ATL prince, maybe it can shed a light on if the marriage would be possible), I think if the bride (Anne of Burgundy) brings with her a good dowry and connections, they can overlook her less than stellar birth (and besides, the pope can always just be asked to make her legitimate, the Grimaldis did it and they were of much lower rank, why couldn't the burgundians do it as well?)