"I've got" is in American English too. "I've got a book." "I have a book." There is no difference in meaning. Only the second one is "standard" American English. Both of them sound fine. Unless I were to think about it, I wouldn't even notice anything "unusual" going on with the grammar.

When 'have' means 'possession' in its broadest sense (ie not when it has another meaning such as in 'Have a walk') there is no difference between 'I have' and 'I've got'. Both are normal in spoken language. 'I've got' isn't used in formal written language, and it's less common in American English. The choice of using one form or the other is entirely involuntary - you'd just use the form which sounds right according to the rhythm and stress of the particular sentence. There is no difference in meaning. In many cases, 'I've got' is actually easier to say. The 'h' sound in 'have' takes some effort to pronounce, and in casual language it's often dropped. 'I've' is a weak unstressed syllable, so we add the 'got' to give it a bit of weight, as if it were a present perfect (like 'I've been' 'I've seen). Of course, it isn't a present perfect, but it's an 'anomaly' grammatically, in that it looks like a present perfect but actually has a present meaning : I have. NB The pronunciation in Manchester is very different from standard GB English in many ways. If you can follow what the actors are saying, you are doing very well indeed!