Purely as an accounting matter, if the market value of your house is greater than what you owe on it, that is “wealth.”
The accounting term for this is "equity", not wealth.
I do not believe there is an accounting term for "wealth". I would be inclined to think of wealth as net worth above a prudent reserve, and above anything earmarked for a specific purpose. But that is a very hard line to draw.
For instance, a college fund for children or grandchildren, is not wealth. OTOH, savings for retirement can be wealth at some point. $10k of savings is not wealth, but $1 million is starting to be, and $5 million definitely is. And, it also depends where it is kept. There are a lot of descriptions for $1 million in a savings account, but wealth is not one that springs to my mind. Foolishness, an invitation to overspend, and a mark ripe for the plucking, all do. That same money as equity in rental property, as a balanced stock portfolio, as working capital in a small business, can all be described as "wealth in the making".
One of the problems here is that vanishingly few newspaper writers have ever had wealth, or even aspired to wealth. Therefore they use the term loosely.
That "aspired to wealth" does not mean sitting on one's rear end, cracking a beer, and saying "It sure would be nice to be a millionaire." It means developing a real financial plan to reaching that goal, sacrificing current consumption for future security, and actually starting to build wealth.
What the article is describing is "net worth", not wealth.
That's what I was wondering about. Median household net worth is already insanely low in the US. If the "wealth" in a house were more realistically computed as the difference between what profit ones current house would bring minus the cost of replacement housing, that would be meaningful wealth. And the median "wealth" would be even lower.
It's hard to imagine a scenerio where anyone except the top 25% (or so) of the population can keep up the illusion of a comfortable middle class life.