When you file taxes, you can take the "standard deduction" that everyone gets. A downward adjustment to the amount of your income that is taxable. Or you can "itemize" your various tax breaks, if they add up to a total greater than the standard deduction.
Historically, home ownership has been "subsidized" in the sense that the interest one pays on mortgage loans is one of the things that can be itemized. And historically, that was usually enough to push you above the standard deduction amount.
But now, the standard deduction amount has been raised to such a high level that only the wealthy and very upper-middle class have any reason to itemize anymore. Typical mortgages now are no longer enough to get you any special treatment on taxes, beyond a renter who also takes the standard deduction.
Of course, this outdated talking point will probably stick around for years, though...
The 'they' you rail against who are amassing properties as massively subsidised wealth accumulators are exactly who this article is arguing against. Deregulate, open up the markets and let people decide on their investments of choice in a competitive environment. Are you really arguing that?