Oskar is the Scandinavian and German spelling of Oscar — which places it in the same name but with a different cultural geography. The K instead of C is a small visual signal that does real work: it tells you the owner either has Scandinavian heritage, has read The Tin Drum by Günter Grass, or simply prefers the Continental spelling of a name they already loved. At rank 1308, it's a name that wears its orthography as a quiet mark of distinction.
The K-Spelling Signal
Name spellings do cultural work even on dogs. Oskar signals something about the owner's aesthetic orientation — toward European taste rather than Anglophone convention. It's a subtle but real difference from Oscar, in the same way that Mikael differs from Michael. Dachshunds and German shepherds whose owners have German or Scandinavian backgrounds are the most natural Oskars. The human name's etymology appears at /names/oscar.
Oskar Schindler and the Literary Oskars
Oskar Schindler from Schindler's List and Oskar Matzerath from Grass's novel are the two heavyweight literary-historical Oskars. The dog-naming context is of course much lighter, but owners who consciously chose the K spelling are usually aware of these anchors and find the depth appropriate rather than burdensome.
Daily Use Reality
In spoken English, Oskar and Oscar are identical. The distinction exists entirely in written form — vet records, dog tags, licensing paperwork. That's either satisfying (you see the correct spelling everywhere) or slightly futile (nobody will ever say it differently). Most Oskar owners are fine with that trade-off, which is a reasonable position.