It depends on what you mean by "saving" and what you intend to do with it.
A couple things to note:
1. A house is extremely illiquid. Plenty of reasons in the posts above me.
2. A house is an investment. Yes, even the house that you live in yourself. It's supposed to appreciate in value, and that's why it's an investment. But it also has its own cost: insurance, fees, taxes, maintenance, etc. So after you subtract appreciation minus these costs, then it *might* be a decent ROI, might not be. But as with most investments, you're taking a risk. Notice that I did not include mortgage in the list of costs, that's because:
3. You have to decouple the money that you used to buy a house, and what you did with that money. Imagine two scenarios:
A: You bought a 1MM house with 300k down payment, so you have 700k mortgage with 4% interest rate
B: You borrowed 700k from a bank with 4% interest rate
These two scenarios are almost identical. The only difference is that in A, you put the 700k into an investment called "house" immediately, whereas in B, you might put it into a portfolio, or whatnot. The other differences are minor: peace of mind, the status of having your own house (even if not paid-off) versus renting, not having to move when your apt lease is not extendable etc.
So if you pay the principal of the mortgage, it is saving in the sense that it's income minus expense. It's a money that you could be throwing away (in exchange of nice clothes / car), but you chose to do something productive with it. So yes it's savings.
But is it always the most prudent thing to put it into mortgage? In scenario B above, would you rather start paying off the bank loan, or would you bolster your portfolio? The answer lies on whether you think your portfolio can gain 4% a year.
The term "home equity" is a myth imo, because once you consider the comparison between two scenarios above, all you're doing is that you're paying off a loan where you could be investing that money somewhere better.