nope.
Phoenix is from ancient Egyptian religion, spread out to middle east and as far as india.
Fenghuang and Phoenix represent different meaning.
Fenghuang is King of the all birds,
and Phoenix doesn't carry that.
While in Chile , they name the capsule "Phoenix" is more lean to the meaning of "rebirth".
It is a miss-translate so ppl mix them all up, also thx to 郭沫若.