The canary is a bird named after an island, and the island was named after dogs.
King Juba II of Mauretania is credited with discovering the islands for the Western world. The Roman writer Pliny the Elder says one island, Canaria, held "vast multitudes of dogs of very large size." It is presumably today's Gran Canaria. The name comes from Latin canis, "dog." The group became the Canariae Insulae, "Islands of the Dogs," and in Spanish the Islas Canarias.
The dogs are not certain. Etymonline says the islands were "supposedly" named for them and suggests this may be folk etymology. Wikipedia offers two alternatives. The "dogs" may have been monk seals, sometimes called sea dogs. Or the name may come from a Berber people of Morocco, called the Canarii in Roman sources. The islands' coat of arms still shows dogs.
The bird took the islands' name, not the reverse. The wild Atlantic canary lives on the Canaries, the Azores and Madeira. It is mostly yellow-green with brownish streaks, not bright yellow. Etymonline traces the English word through French canarie to Spanish canario, "of the Canary Islands." English had "Canary-bird" by the 1570s and shortened it to canary in the 1650s. The islands' wine was already called canary from the 1580s.
The birds spread through Europe as a trade. Wikipedia says Spanish or Portuguese sailors may have brought them over in the late 14th or early 15th century. Spain protected its business by selling only males abroad and banning the export of females. Monasteries profited from the trade, and monks bred the birds. Canaries were first bred in captivity in the 17th century.
Breeding is where the famous yellow came from. A mutation suppressed the dark pigments of the greenish wild bird, producing the bright domestic canary. The color name canary yellow comes from that bird. So a shade of yellow is named for a bird, named for islands, named for dogs.
The bird then found its grimmest job. After investigating a mine explosion, the physiologist John Scott Haldane proposed using animals more sensitive to poison gas than people. Mice were tried first, but canaries proved more sensitive and showed distress more clearly. Their use in mines is documented from around 1900. Britain phased them out in 1986 in favor of electronic detectors.