In English, we would generally use this in the past tense. because at the time it is happening i.e. the present, you still think the other person is coming, just running late. So we'd say 'I got stood up.'
"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.
Leo cây = to climb a tree right?
Got stood up = I was given the opportunity to climb tree (because my friends didn't show up)?
Or completely unrelated to that verb?
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
Does this mean 'stood up' as in - go on a romantic date and the other person doesn't show up? Or as in - fall over and be righted?
The former, but this doesn't necessarily apply to romantic dates only. Say you have an appointment to hang out with a friend but they don't show up, then this could also be used.
Did you hear about Stepen Hawkin's first date? He came back with scuffed knees and hands and his glasses all broken.... Yup, she stood him up.