From early times to the 1500s, the Scandinavian countries were loosely united, either as Vikings or as the Kalmar Union. From the 1500s to the Napoleonic Wars, Denmark and Norway were one country. Sweden effectively annexed Norway and the two made a single country for nearly a century.

Even today, Danish, Norwegian, and Swedish are mutually intelligible languages. The countries share a great deal of political and cultural ties. It's not at all a stretch to think that you had ancestors who more closely aligned genetically with one country but found themselves living in a different one. That goes as well for the Germanic Europe portion of your test--there could have easily been cross-border movement from what's now Germany into Denmark, with the people eventually living as Danes but remaining ethnically German.

To use another example from my wife's homeland--there are hundreds of thousands of Chinese who have moved over centuries from China to Vietnam. However, relations between the two have never been better than semi-cordial and are usually more frosty (including today). As a result, nearly all of them "Vietnamise" their names--both family and given--and live as Vietnamese people. Some of them retain their Chinese heritage behind closed doors, while others lose it and fully assimilate. But if you gave them an ancestry test, it would come out as Chinese (plus whatever amount of Vietnamese they intermingled with).

Human migration makes DNA messy. The results sound completely plausible and really shouldn't shake your own view of yourself. If your grandma came from Denmark and thought of herself as Danish, then that's more what you are than what your DNA says.