Buying has costs too. Mortgages are an obvious example, but even if you could afford the house without debt the money could also have been invested in something that yields income like stocks or bonds. If the amount you'd receive in income from investing the house price can cover the rent of an equivalent house, it can make sense to rent as you come out ahead overall. It does not surprise me at all that rich people (who can take the time to shop around because they are not at risk of getting evicted) manage to get a better deal than someone who represents more credit risk for the landlord.
Of course pulling this type of thing off requires that you can find investments that will yield more than the rent price, which may not be easy to do in the long term as high yielding investments can often carry unforeseen risks. Also because a significant amount of people already know this, when renting is "cheap" enough compared to buying the demand for renting will increase and prices will rise. This means that the condition is unstable and will often disappear on its own.
Still, there are sometimes very valid monetary reasons to rent instead of buy. Then there are nonmonetary reasons to rent instead of buy that make sense even for quite rich people. If you're a NYC banker and you get seconded to London for 2 years to set up some new division, it is probably not worth the hassle to buy a house there.