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Old English sceadd "shad," important food fish in the Atlantic, possibly from Scandinavian (cf. Norwegian dialectal skadd "small whitefish"); but cf. Welsh ysgadan (plural), Irish and Gaelic sgadan "herring." OED says Low German schade may be from English.
Its importance suggested by its use in forming the common names of U.S. East Coast plants and wildlife whose active period coincides with the running of the shad up rivers, e.g. shad-bird, shad-bush, shad-flower, shad-fly, shad-frog. From the shape of the fish comes shad-bellied, 1832 in reference to persons, "having little abdominal protuberance;" of coats (1842) "sloping apart in front, cut away," especially in reference to the characteristic garb of male Quakers.