Compare it with group of Robert Ferguson photos that I've already posted. (I re-posted one of them below so you really won't have to go to any trouble whatsoever.) You'll note that the odd white "mountain" appears to be the same in both views, as does the bridge (reflected in the water in the old photographs). And you can see a stone lantern that looks just like the ones in the photos.
But what you DON'T see are the crazy trees, the dog statue, and the weird rock-covered iron grate that the woman was sitting on (though maybe that's too small to show up in the postcard).
The Japanese Gardens were in Overton Park for some 40 years. My theory is that this place, like so many others around town, changed over the years. Certain features were added or removed.
But it's pretty definitely the same "mountain," if you ask me.