I think Minato absorbed the 9-tails' power, and has been living using a transformation jutsu, disguised as his supposed 'son'. Jiraiya's deathbed memories of a pregnant Kushina were purely electric impulses in his dying brain, combined with the chakra-scrambling effects of the six-paths-of-LSD-swords imbedded in his spine, and hold no truth. Kushina never existed, which is why she has not been seen in the story.
Similarly, Lee is the bastard son of Gai and his kage-bunshin, born from a test tube. And he knows this.