Is a Phoenix the same as an archaeopteryx?

No, an Archaeopteryx is a prehistoric bird, while a Phoenix is a mythical bird. Archaeopteryx was not discovered until 1861, while the myth of Phoenix was referred to as early as 170AD. Those who would try to link the two in any way will also be disappointed, as Phoenix was described as very large, while the largest of the Archaeopteryx specimens was only about the size of a magpie.