The Subway Stamp Shop hinges were "Dennis's" hinges, I think. Named after one of their grandsons. And it sounded enough like the famous "Dennison's" hinges to remind buyers of them. I'm not sure if that brand is still sold. It was supposed to duplicate the old, very popular Dennison's hinges, but it never did for some unknown reason(s).
I've never heard a good explanation of why Subway Stamp could not duplicate the original Dennison's hinges. But yes they did buy the old Dennison hinge-making equipment, probably a single machine of some kind. So they did try, but I've never heard anyone explain why Subway could not get the right ingredients to make the proper Dennison-style glue -- if that was in fact the problem -- but I suppose that makes sense. Everything I have heard is pure speculation of the "I've always heard" variety. Subway had the Dennison machine, so their problem must have been something else like the adhesive. I can't imagine the glassine paper made much difference, but maybe. Glassine is readily available, isn't it? So getting the formula right for the adhesive seems likely to be the culprit.
But then there's the larger mystery of why no one can duplicate the original Dennison's hinges anymore. That nearly always points to the adhesive as the cause, as well. The "horse glue" claim is fairly popular. If animal glue was the key to Dennison's removable adhesive, but it got banned, that would explain the demise of these hinges. The problem is I've never seen any evidence to prove the "horse glue" claim, just speculation. But it also could be connected to waning sales of stamp hinges, as some have suggested.
Some people say it was the glassine. Others have claimed it wasn't the glue itself, but the way Dennison applied the glue in a kind of checkered pattern made it less aggressive. But both of these are also just speculation.
Finally (I hope), there's the additional mystery of why Dennison "destroyed its records" about the glue formulation -- if that's what they did. If they did destroy records, as many people claim, why? Or maybe the company just threw away old files? Maybe it wasn't as dramatic as "destroying the glue formula," as some have claimed?
I guess you pick your theory as you think best. This is how conspiracy theories start, isn't it?

Today, modern companies like 3M sell products like adhesive tapes with "removable adhesives", but maybe they have little incentive to make stamp hinges? It wouldn't be difficult to engineer removable tape or 'sticky notes" into hinges, but who knows what modern removable adhesives will do to paper and stamps over long periods of time? I guess we can experiment and see what happens. On cheap stamps, of course!