c. 1600, "flaps formed by the lower back of a coat," from coat (n.) + tail (n.). In 17c., to do something on one's own coattail meant "at one's own expense." Meaning "power of one person," especially in politics, is at least from 1848 (in a Congressional speech by Abraham Lincoln); expression riding (someone's) coattails into political office is from 1949.
But the gentleman from Georgia further says we [Whigs] have deserted all our principles, and taken shelter under General Taylor's military coat-tail, and he seems to think this is exceedingly degrading. Well, as his faith is, so be it unto him. But can he remember no other military coat-tail under which a certain other party have been sheltering for near a quarter of a century? Has he no acquaintance with the ample military coat-tail of General Jackson? Does he not know that his own party have run the five last presidential races under that coat-tail? And that they are now running the sixth under the same cover? Yes, sir, that coat-tail was used not only for General Jackson himself, but has been clung to, with the grip of death, by every Democratic candidate since. [Lincoln, speech in Congress, July 27, 1848]