Whether it is true or not, a story is that George VI sent a memo to the King of Denmark stating something to the effect of "If a Bowes-Lyon is good enough for me, it should be good enough for one of your minor princes."
Regardless of the veracity of the above, it would have been an insult to Queen Elizabeth to give her niece's marriage a morganatic status.
Princess Anne was very conscious of her lineage and was something of a snob. Quite the opposite from her gracious sister-in-law Countess Ruth af Rosenborg.