Tom at Huntercombe all competitive matches played by the club are foursomes - one round in the morning (very, very seldom longer than three hours) followed by boozy lunch and the same again after lunch. This is the format for the vast majority of competitive golf between traditional clubs in the UK.
Additionally, all the competitive tournaments within the club that involve partners are played in the foursomes format.
As Sean points out, it has its origins in the fact that most traditional competitive golf in the UK is played in winter, so getting round quickly is a big deal. The best example (albeit not foursomes) is the Presidents Putter always played in freezing January weather.
I think it is a great way to play golf - and demanding too. If you are off form, it is difficult to get into the groove because gaps between similar shots can be long. But by far the most demanding issue is dealing with your partner - not putting them in trouble, not getting down when you do so, and - above all - never, ever being judgemental.
Re mixed foursomes, it is sometimes affectionately known by husband/wife partners as "divorcesomes"! But your anecdote suggests it works both ways!
As for funny stories: when I still lived in SA the champion club golfer for many years and a celebrated amateur, and humourist, was a man called Reg Taylor. The 11th hole at the River Club is a c 400 yard par 4, with a downhill carry to water of about 300 yards. The second shot was typically a shortish 100-120 yards, but all carry over water, so daunting for many golfers.
One day Reg was playing mixed foursomes and hit a good drive down close to the water. When his partner asked him what she should do, Reg replied "For god's sake....miss it!".