"I get stood up" means this happens to me with some frequency. "Am being stood up," or "am stood up" or "am getting stood up" would apply to a singular ongoing situation. If an English speaker leaving the scene of the event uses the present tense, he DOES NOT say, "I get stood up." Granted, he is more likely to use the past tense.
It is unrelated. This is one of those slangs that no one really knows the origin of, and depends largely on context to determine the meaning.
Someone once explained to me that Viet couples in the past used to choose a tree as their rendezvous, and when one person is late, the other might climb it to be able to see if their date is coming from afar. That's the most plausible explanation I believe in tbh