The form is the same as the present perfect, but the meaning refers only to present time.
Originally, a statement such as "I've got a problem" did refer to a present perfect concept ( the idea that you have acquired a problem in some way) . But over time, we've come to use the 'have got' form as an informal/colloquial alternative to 'I have', and the idea of acquisition has been lost.
Just think of it as an idiomatic way of talking about a present situation.
By the way, it is always worth remembering that the present perfect is a PRESENT tense, because it tells you something about the present time. Just as "I've lost my key" tells you that you don't have your key now, "I've got a problem", tells you that you have a problem now. If you look at it this way, it isn't that strange, is it?