"Lady Lazarus" is full of fun times, huh? Oh wait. The big, controlling metaphor of the poem is Lady Lazarus's comparison of herself with the fate of the Jews who were slaughtered (all six million of them) during the Holocaust. She compares her emotional suffering to their physical suffering (and deaths). She takes on their suffering to explain their own. Obviously, this daring move makes us ask a whole lot of questions, but it's also a reminder that every line of this poem is going to be majorly depressing. So have a rom-com ready for when you're done.
Lady Lazarus's suffering is nobody's fault, but the Holocaust was perpetrated by a specific group of people—the Nazis—so it's not a fair comparison to make.
The Holocaust metaphors in this poem are so extreme that they actually make Lady L's suffering seem pathetically minor in comparison.