This article is discussing an income tax, not a wealth tax, which you correctly state would not be legal at a federal level without an amendment.
It's hard to really interpret what the article is actually talking about though due to a lack of specifics. Billionaires do pay quite a lot of taxes (the top 1% of Americans pay 43% of all income tax revenue, for instance), and also pay a lot in consumption taxes, but due to the way their income is structured, a lot falls under capital gains and other taxes rather than standard tax on salary. Gains in assets contribute to net worth, but (correctly) are not taxed until those gains are actually realized. There are some loopholes in how those assets can be used as collateral for loans, but since those do still have to be paid back with interest, they do quite literally not represent net income. There's a difference between aiming to close tax loopholes and simply trying to get more revenue from billionaires. Neither is necessarily a bad aim, but they're not exactly the same thing. There's also an important distinction to be made between taxing billionaires because they're a useful source of revenue for important government programs and taxing them because you just think they're icky and shouldn't exist.