It's easy to get a statistical distribution that does not have finite variance. For example, you can sometimes get that when the thing that is measured comes from two random variables, one divided by the other. Sometimes you can't measure a variance. Sometimes you can get a statistical distribution that doesn't even have a mean.

When that happens, if you don't notice, you can get a mean and standard distribution from the data. And when you collect more data it will seem to mostly fit. But you get big outliers more often than you'd expect. As you recompute your mean and standard deviation, with more data the standard deviation keeps increasing. Because the longer you keep measuring, the more unexpected events you will have that go outside the predicted range.

Without knowing much at all about floods, they seem to fit this pattern. The news keeps announcing floods that were supposed to be unlikely, unexpected.

Of course, for all I know this is just the news reporting things that should have been expected. Of all the thousands of places we calculate hundred-year-flood levels for, every year we should expect floods that high at one percent of them. Maybe what's happening is really exactly what should be expected.

But it's testable. With enough data you can check whether flooding ought to fit a finite-variance distribution or not.

How well has it been tested?

As far as your question regarding finite variance, my understanding is that the LP III distribution asymptotically approaches unity and thus remains unbounded without finite variance since from a risk perspective there could always be a larger flood that has not yet been measured. There are also methods for estimating a Probable Maximum Flood (PMF) that is meant to represent the most extreme combination of meteorological and hydrologic conditions that are reasonably possible in a catchment.