I have a friend who has exactly the same accidents, injuries, mess ups, failings, etc., as I but I can't find a word in English (or otherwise) for it. Physically, we're opposite as night and day. But when something goes wrong, I can match story for story.
Assuming that the absolutely-not-English word "shlemazlut" is derived from the Yiddish word "schlemazel" to mean the foolery of such a fool as a "schlemazel," then the word you're probably looking for is "schlemiel."
In Yiddish, the counterpart of a "schlemazel" is a "schlemiel." They aren't identical twins but instead are two very different fools, two very different fools engaged in the same foolery and two very different fools who come together in a complementary fashion as a dyad such that in their ongoing foolery, the "schlemiel" becomes the one who spills the soup and the "schlemazel" becomes the one who slips on it.
I use the term "mirrors" when it comes to person who exhibit similar thoughts "whoa they completed my sentence..."
also, is your ask neologistic too? leaving aside your tag of "single-word-requests": The thought "quantum entanglement" comes to my mind, and hence I posted this question: https://physics.stackexchange.com/questions/503894/noun-for-particles-that-are-quantum-ly-entangled
so until that is answered ... "he/she is my quangler / quantum-mate / quantum entangler" :p
Edit: as per answer there, "entangled" can be a word that will connote the meaning you want to convey when the audience is bit sciency
Partner in misfortune or a broader synonym fellow-sufferer could fit there.
The skate's partner in misfortune is the angel shark, also known as the monkfish.
Amos Oz's A tale of love and darkness also uses the phrase. It's the English translation of the Hebrew, which made me wonder if the original also used some phrase with mazal:
Often each sees in the other not a partner in misfortune but in fact the image of their common oppressor.