No ancient reader treated the Metamorphoses as religious authority. Roman religion had no scripture and located its authority in inherited rites, priestly colleges and state observance, not in books about gods; a poem retelling myths was entertainment and education. Its enormous influence is a matter of transmission, not sanctity.
That transmission survived the end of paganism because medieval readers found a way to keep reading it. Ovid was copied through the monastic centuries and became so widely studied in the twelfth and thirteenth that the period is sometimes called the Ovidian age. The characteristic device was allegory: the Ovide moralise, an anonymous French verse work of the early fourteenth century running to some 72,000 lines, retold every story and then supplied a Christian moral for it, so that pagan transformations became figures of doctrine. Comparable Latin commentaries did the same. The readings look strained now, but they are the reason a poem of pagan myth stayed in circulation.
In English, Arthur Golding published a complete verse translation in 1567, in fourteen-syllable couplets, the first made directly from the Latin. This was the Ovid Shakespeare knew, and the borrowings are extensive: Venus and Adonis and the Pyramus and Thisbe farce in A Midsummer Night's Dream come straight from it, and Prospero's speech renouncing his art in The Tempest follows Medea's incantation closely enough to be traced line by line. Through Golding, and through painters from Titian to Bernini working from the same stories, the Metamorphoses became the default form in which classical myth is known.