So when I think I heard an American said “I got (something)”, actually he/she said “’ve” after the “I” and it’s present perfect? And this is the time where present perfect should be used? And then it's not really possession but is more about acquirement, right?

(I’ve learned from Swan’s Practical English Usage (237.5-7) that it’s British English that says 'have got' and that is double form of have and means just exactly ‘have’ to mean possession and this is not present perfect but present tense. (No wonder I’ve been confused..))