Phoenixes are referenced through multiple Eastern cultures and the Bible. Through some force the myth or what the 'bird' represented made it into the new world. It's still a symbol used these days.
The bird part though... why does it have to look like a bird? Think of what we saw with the dragons. What we're seeing with the kitsune boy running around. The dragons weren't exactly giant winged lizards. The kitsune isn't exactly a multi-tailed fox. There's an animalistic part to both creatures, yes, but they appear human enough.
What if the phoenix wasn't a bird. What if it could assume a human form and blend in? They hail from places like Egypt and Lebanon. They'd prefer a warmer climate, something similiar to what they're used to. If they made it into the new world along with the mythos maybe we're not digging in the right places.
Then again, who knows? This is all just me throwing out crazy ideas.