There are default, general rules on stacking “bonuses” as well as combining magical effects that mean that usually, recasting the same spell does nothing more than refresh the duration and provide redundancy in the face of dispel magic. These don’t block fear effects from “stacking,” however:
Fear conditions aren’t “bonuses,” though the conditions—not the effect itself—may cause them.
The combining magical effects rules say that “Except in special cases, a spell does not affect the way another spell operates,” but here we are talking about a special case—this is known as the cumulative nature of fear. Officially, things wind up under the “Same Effect More than Once in Different Strengths” rule.
Specifically, when something says “X becomes shaken” when X is already shaken, X should become frightened, instead. As you note, many sources of fear specify that you cannot combine them with themselves—but not all. For example, perhaps the quintessential example—cause fear—does not. This makes some sense because to apply cause fear again, you have to burn a spell slot again. And we’re talking about “Same Effect More than Once in Different Strengths” because the second cause fear is causing the frightened condition, not the shaken condition.
All of that said, it seems to me that very often authors and editors forgot about this. It is extremely common for effects to be written seemingly expecting that they won’t stack due to the “same source” rule, even though they aren’t actually “bonuses”—the authors seem to forget that those rules only apply to bonuses and treat it as the default in general. Likewise, while many cases specify that a fear effect can’t stack with itself, it’s not clear to me that every author always understood that a fear effect must specify that if it’s not to stack with itself—because pretty much everything that isn’t fear won’t stack with itself like that. It is also very common to find tables where you need separate effects to ramp up fear, which is likely true of the tables for at least some authors.
Ultimately, published authors should be better than that—and Paizo’s editors should definitely be better than that. But the Deadhand Style feat is already known to be poorly written—your other question about its duration proves that, because I’m pretty sure there just isn’t any answer to it. And unlike cause fear, where you have to burn more spell slots to ramp up the fear, if Deadhand Style’s empowering of your fists lasts for multiple attacks, you get the opportunity to ramp things up without burning more resources. That might indicate that it’s not supposed to stack with itself—or, for that matter, it might be a hint that it’s not supposed to last more than one attack in the first place. Who knows?
In the end, I’d recommend just... pretending that the Deadhand Style feat doesn’t exist. It’s cool, no doubt, but the questions around it are really problematic. And it’s not that strong a feat—fear immunity is widespread and desirable, so a lot of targets are going to be avoiding the effect altogether, including entire creature types. And for that matter, the (original, “chained”) monk class itself is very weak—though, of course, there are other ways to get a ki pool that are better, such as the ninja or, ya know, the monk unchained.