The Strait is way far from Denmark, so why is it called the Denmark Strait?
Greenland is owned by Denmark.
And up to 1948, Iceland. For various definitons of “owned”.
Because Denmark explored it. When the Union of Kalmar broke up in the 1520’s, Denmark inherited sovereignty over Norway and Iceland. It also inherited the Norwegian claim of sovereignty over Greenland, based on the old Norse settlements, but this claim was in abeyance as the Norse settlements had died out.
The hyper-ambitious Danish King Christian IV led a Danish naval revival starting around 1600, and as part of this revival he sent a series of expeditions to Greenland, attempting to reassert sovereignty and determine the fate of the Norse. The expeditions failed in the short run, as they neither found the Norse nor established a successor colony, but they poked around the Denmark Strait quite a bit, and named it, and Denmark did eventually recolonize Greenland in the Eighteenth Century.
And places often, but do not always, keep their names even when sovereignty is transferred. When the Dutch settlement of Nieuw-Amsterdam was taken over by the British, it was renamed New York after the Duke of York. However, after US independence, we didn’t find sufficient reason to rename it again. Why didn’t we rename it “Americatown Island” or “Independenceville-by-the-Hudson”? Or, why does New Orleans still have a “French Quarter”? Why not rename it the “Reagan Corner”? Or “Eisenhowersburg”? "How about “Barack Square”? And don’t get me started about the name “Louisiana” itself. Why are we naming a place after a French king? He’s not our buddy, guy.
Why they changed it, (or didn’t,) I can’t say. People just liked it better that way? 
Right, that’s the point. So why do people still call it the Denmark Strait? Maybe people just like to call it that, or at least they can’t be bothered enough to take action to get it renamed. So that’s its name.
Well, in my opinion, it’s nobody’s business but the Yanks.
mmm
But he is a legend.
FWIW, though it’s a bit off topic, there’s a river in Northern California that was variously called the Salvianka and Rio San Jacinto, but today is known as the Russian River. Russian and Slavianka becuase there were at one time Russian colonists not too far from there at what today is called Fort Ross. I think the original colonists called it Krepost Rossiya or Krepost Ross’ (Fort Russia).