According to HMRC their income is actually in the top 99th percentile, not just the top five. I used to be on an income slightly in excess of that of the featured household, and it was NOT a financial struggle, even with one kid at private school, no child benefit, and hefty pension contributions. However, we didn't eat out, didn't have fancy holidays, and ran an old car. Also, we bought our house 20 years ago, so have paid off the mortgage. Our priorities were childcare costs (about £40,000 p.a. after tax, for three kids - two at state primary), paying off debt, and saving. In consequence I could afford to retire early, and the youngest kids have remained at state schools.
So it's a question of choices. The point of the article is that the traditional middle class - lawyers, doctors & other professionals - can no longer afford all the paraphernalia they consider their income and standing entitles them to. It's been an old news item for years that private education is moving beyond the reach of the traditional middle class (to single out one expensive choice).
The societal interest in the story is that the traditional middle class has always been considered the bedrock of an advanced, progressive and financially well developed society, and there is a concern (perhaps only philosophical) that the erosion of the middle class will lead to - or is a sign of - economic erosion, and a return to the world of some very rich folks, and millions of poor. That is not generally thought to be a good idea.
However, with the exception of private education, where fee increases have always outpaced inflation (something to do with parents wanting 'state-of-the-art' facilities perhaps) it seems to me that the impoverishment of the middle class has more to do with the greater availability of luxury goods and services (fancy cars, fancy holidays, ridiculous kitchens) which means a 'middle class income' has to (its recipients would like it to) stretch even further. But these are choices.