Eh? Standard American and British usage is the same with regard to quotes that span multiple paragraphs. Given that it's understood that speakers can alternate without each quote being attributed, e.g.:
Bob said: "Any opinion on this, John?"
John said: "I have two things to say."
"What are they?"
"One of these things is this.
"The other thing is this."
– how would you punctuate that? If you close each paragraph with a quote, then there's no way to tell who's speaking except to label each paragraph:
Bob said: "Any opinion on this, John?"
John said: "I have two things to say."
Bob asked: "What are they?"
John answered: "One of these things is this."
John continued: "The other thing is this."
And if you don't open each quoted paragraph with a quote, it's very hard to tell which paragraphs are quoted:
John said: "I have two things to say.
One of these things is this.
The other thing is this."
The third thing, he kept to himself.