It isn't just British people. For example, a popular TV quiz show in the United States in the 1960s was called "I've Got a Secret." I don't believe there is any good logical or grammatical explanation. A scholar might be able to find some explanation in the history of the language.

Just a note that it's not just British people. This pretty common in North American English as well, just perhaps a little less common.

Great question. "Have got" is just a more informal version of "have", but they are identical. "How many sisters have you got?" = "how many sisters do you have?" "I have got a cold" = "I have a cold" "I have got three bikes" = "I have three bikes". It is worth noting that you can only use the "I have got" construction in the present tense or future perfect ("I will have got..." e.g. "Have you got the tickets yet?" "No, but I will have got them by tomorrow/I will have them by tomorrow")

Hi, if you’re interested in working on your pronunciation for improved communication, you’re welcome to contact me. I’ll be offering pronunciation training on Italki; if this is something that you might be interested in, feel free to get in touch. I’m a qualified teacher with a British accent. Adriane