The phoenix entered Christian use very early and very explicitly. Clement of Rome, writing to the Corinthians around 96 CE, cites the bird by name as a natural sign that God provides of the resurrection to come, describing the death, the worm born from the decaying flesh, and the flight to Heliopolis. He offers it as evidence. Later writers, Tertullian, Origen, Ambrose, take it up, and the phoenix becomes a standard motif in catacomb painting, mosaic and funerary sculpture, usually shown with a rayed nimbus.
It is worth being clear that these writers believed the bird existed. The phoenix was not for them a metaphor but a fact of natural history that happened to be theologically convenient, in the same way that the peacock's supposedly incorruptible flesh was for Augustine. The shift from creature to symbol happens later, and it happens because the natural history collapsed.
Two other birds are routinely called phoenixes in English and are unrelated creatures.
The Chinese fenghuang is an auspicious composite bird, paired with the dragon, associated with the empress as the dragon is with the emperor, and with the arrival of peace and virtuous rule. It does not burn, does not die, and is not reborn. The translation as phoenix is a European convenience of the early modern period and it has misled generations of readers.
The Persian simurgh is a vast benevolent bird of the Shahnameh who raises the abandoned Zal, and in Attar's Conference of the Birds is the goal of a pilgrimage whose resolution turns on a pun. The Slavic firebird is another creature again. None of these is the Mediterranean phoenix, and the resemblances that exist are resemblances of function rather than of descent.