Tldr answer: Advanced power armor is the finalized result of the X-01 project.
Full answer: The U.S. military was developing a new suit of power armor before the war designated the X-01. It was only a prototype and had not completed testing when the bombs dropped, but the Enclave had copies of its designs at the oil rig. They completed the project, creating the advanced power armor mk1. Elsewhere in the Wasteland surviving X-01 suits were used by other Enclave survivors and military remnants.
It's a confusing topic because Fallout 4 never explained the reason why X-01 and APA were almost identical, leading to the fan theory they were the same. We eventually got the answer in 76.