Dictionaries and other reference books don't have the power to mold the world to conform to the desires of us tea lovers. They have to explain to people working with tannic acid every day such things as its applications, which include, according to Merck 13 (entry #9141):
"Mordant in dyeing; manuf ink; sizing paper and silk; printing fabrics; with gelatin and albumin for manuf of imitation horn and tortoise shell; tanning; clarifying beer or wine; in photography; as coagulant in rubber manuf; manuf gallic acid and pyrogallol; as reagent in analytical chemistry.
"THERAP CAT: Astringent.
"THERAP CAT (VET): Astringent, hemostatic, in solutions for burns. Has been used internally as an astringent and as a heavy metal antidote."
Leather tanning is a trade relegated to the lowly burakumin in Japan (who also handle garbage and corpses) and is banished to districts far downwind from human settlements in the Maghreb. I don't know much about it except that it requires letting the skins of dead animals decay somewhat before treating them with lye and tannic acid, and that it smells very bad.
I doubt there's much tannic acid in tea, especially since the Merck article contains a reference to a study of its toxicity and says in italics, "Keep well closed and protected from light." It also says it's "produced from Turkish or Chinese nutgall." So tea is obviously not the preferred source of it.
(Incidentally, nutrition fans, it occurs in the bark of myrobalan. Doesn't that ring a bell?)