I'd say that 13th age is closer to 4e than PF2 is, but you can definitely see some 4e stuff in PF2. I think this is primarily a case of convergent evolution. PF1 is basically 3.5e with More Stuff, so the problems of PF1 are basically the same as 3.5e. So it makes sense that in some cases they would come to similar solutions.
I can see the argument, though 13th Age is a little--minimalist?--in some ways for the effect to be as visible there. You're correct, of course, that similar problems lead to similar solutions.
Take monster design, for example. 3e used, at least in theory, "organic" rules for designing monsters. Each monster type was basically the same as a (bad) PC class, so HD translated into not only hit points but also attack, save, and skill bonuses. Then stat bonuses and equipment/natural armor was added on top of that, and eventually some actual "runtime" numbers emerged, and in theory you used those plus special abilities to assign a CR to the monster. Except (a) that was really complicated, and left a lot of room for error, and (b) it would often get you monsters with either ridiculous peaks or glaring weaknesses, which was part of what made the CR system a joke (hello Mr. CR 9 frost giant with Will +6). In practice, good designers would have an intuition or if they're lucky formal benchmarks regarding what final stats are appropriate at various CRs, and then massage the stats appropriately. But that could sometimes lead to weirdness like boosting Dex to get a decent Reflex save, and as a consequence ending up with Initiative +8 or something. Plus, there was no formal guidance on what stats were appropriate, just trial and error. And there's very little material around for the public discussing the issue (the one product I can think of is Trailblazer, which was a 3rd party product released near the release of 4e that basically tried to backport a lot of the 4e ideas into 3e).
Yeah. I'll tell the truth: I'm not a massive fan of the way 4e and post D&D games treat NPCs and PCs as different as they do; I come from games like RQ and others where that's just not a thing, and it feels kind of weird and artificial to me.
But having run 3.5 D&D up through 14th level, I can't say I don't understand it; in addition to making CR a joke (though it wasn't the only reason; having so much ability to bake a cake on a PC didn't help) it made higher level opponents of some types almost impossible to run (because there were so many moving parts).
The obvious solution was of course to start with CR (or level), and set stats based on benchmarks instead. Now, the two systems diverge somewhat on the details (4e has monster roles that set these benchmarks, while PF2 has level provide a range and its up to the designer to make sure that the creature has strengths and weaknesses within those level-based bounds), but they're based on the same principle.
And of course it doesn't hurt that Paizo has one of the more prolific 4e designers, Logan Bonner, on staff. But if I were to hazard a guess, it would be that his role would be to guide away from how things were done in 4e when they didn't work, and perhaps come up with a better solution with hindsight. That is, not to go "Oh, in 4e we solved that problem in this way, so let's copy that" and instead more "In 4e we solved that problem like this, but that in turn caused these issues, so what if we try that way instead?"
Well, its entirely possible in a few cases it was "In 4e we addressed this in this fashion, and I still haven't seen a better solution so far, so maybe we at least want to inform what we do on it." I wasn't a big fan of 4e, but to not say it did some things right would be a--take.