"我管你张三还是李四" literally means "I care you are John, Dick or Harry". And the actual meaning is "我(不)管你张三还是李四" meaning "I (don't) care you are John, Dick or Harry"
Using the opposite term "管" (care) instead of the actual term "不管" (don't care) is a sarcastic way of making a statement.
What common variants on this phrase exist where the surname stays the same?
It is not a set phrase, you can substitute any names/nouns to indicate "I don't care who you are". For example: "我(不)管你是皇帝还是王八", "我(不)管你成龍还是成蟲"
When someone said he is the emperor but you don't care, you can say "我管你皇帝还是王八"
When Jackie Chan demands you to do something because he is Jackie Chan, and you don't want to do it, and don't care it is Jackie who demanded it, you can say "我管你成龍还是成蟲"
One more: "我管你特朗普还是特朗通，你是个混蛋" roughly translated as "I don't care you are Donald Trump or Donald Dumb, you are an asshole"