I can not give you a clear answer, but two things come to mind. Karen's husband, Bror von Blixen-Finecke, whom she had married in 1914, was a step-grandson of Queen Alexandra's maternal aunt Princess Augusta of Hessen-Kassel.
Queen Alexandra and Karen's mutual connections to the noble Krag-Juel-Vind-Frijs family might well have played a role, too. Karen and Bror were both closely related to this family, which was one of the richest and most prominent in the Danish aristocracy. In 1916, Bror's uncle, Count Mogens K-J-V-F, would have been head of this family who resided at Frijsenborg manor. Karen was a close friend of his daughters, who were also her own second cousins. Mogens K-J-V-F was married to Countess Fritze Danneskiold-Samsøe, whose mother was almost like a sister to Queen Alexandra, as she had lived with the royal family in Copenhagen as a child. Furthermore, King Edward was a friend of Mogens K-J-V-F, and while still Prince of Wales, he and Alexandra paid a visit to Frijsenborg in 1897. In 1913 Queen Alexandra returned to Frijsenborg with her sister Dagmar.
A photo of Karen Blixen (right) with her close friend and second cousin Countess Daisy Krag-Juel-Vind-Frijs a few years before she (Daisy) took her own life:
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