Tintorería means both dry-cleaner and place for dying clothes. The latter is the older definition, and follows the words etymology. (See it here, but it is clear from tinturar's meaning to dye.)
To dye clothes and to clean clothes seem to me near opposites, but I do understand that both might be performed by the same person—a person who works with clothes. Tintorería, though, doesn't refer to all clothes washing, but only to dry clearning. Besides, other occupations also relate to clothes, like a tailor, and aren't included in tintorería. Tintorería did not become a general term for places that deal in the after-sales clothes market.
Why, then, does tintorería mean dry-cleaner?