"flat end of an arm of an anchor," 1560s, perhaps from fluke (n.3) on resemblance of shape, or from Low German flügel "wing." Meaning "whale's tail" (in plural, flukes) is 1725.
"flatfish," Old English floc "flatfish," related to Old Norse floke "flatfish," flak "disk, floe" (see flake (n.)). The parasite worm (1660s) so called from resemblance of shape.