"weathercock," late 14c., from Old English fana, fona "flag, banner," from Proto-Germanic *fanan- (source also of Old Frisian fana, Gothic fana "piece of cloth," Old High German fano, German Fahne "flag, standard"); possibly cognate with Latin pannus "piece of cloth" (see pane).
Old English fang "prey, spoils, plunder, booty; a seizing or taking," from gefangen, strong past participle of fon "seize, take, capture," from Proto-Germanic *fāhanan (source also of Old Frisian fangia, Middle Dutch and Dutch vangen, Old Norse fanga, German fangen, Gothic fahan), from nasalized form of PIE root *pag- "to fasten" (source also of Latin pax "peace").
The sense of "canine tooth" (1550s) was not in Middle English and probably developed from Old English fengtoð, literally "catching- or grasping-tooth." Compare German Fangzahn. Transferred to the venom tooth of a serpent, etc., by 1800.