In your examples, “got it” is not the simple past tense. To form a question in the simple past, we would say “did you get it?” Even if we drop the “did you,” it’s still “get it?” in the past tense (“did” is understood). In your example, “got it” is short for “have you got it?” This is simply a common idiomatic expression with a (present tense) meaning similar to “do you have it?” Alternatively, this could be considered a form of the present perfect. We use it when there’s a connection between the past and present. Your friend has already understood (past), so now he is going to go (present / future) pick up his shirt.