Etymology of "wearing thin"
The connotation of wear of “using up, consume” is quite old:
Wear:
Secondary sense of "use up, gradually damage" (late 13c.) is from effect of continued use on clothes.
(Etymonline)
Its figurative usage is more recent, especially the second figurative sense of wear thin:
- Be weakened or diminished gradually, as in My patience is wearing thin. (Late 1800s)
- Become less convincing, acceptable, or popular, as in His excuses are wearing thin. (First half of 1990s) Both usages transfer the thinning of a physical object, such as cloth, to nonmaterial characteristics.
(The American Heritage Dictionary)
Note that Google Books usage examples of wear thin from 18th c. and most of 19th c. are in the physical sense.