Actually it boils down to the “stock link” in the upper right hand corner on the play framework home page which links it to http://www.reactivemanifesto.org/ - and the fact that the Phoenix home page doesn’t brandish the same link.
While the Play framework may be dealing with the exact issues that you mention there is no requirement in the “Reactive Manifesto” that “reactive systems” have to be dealing with the issues on exactly those terms (which is why I personally find the term “Reactive Manifesto” misleading). The manifesto is really about systems that are in general responsive, resilient, elastic, and message-driven. @OvermindDL1’s analysis comes to the conclusion that Phoenix lends itself to building responsive, resilient, elastic and message driven systems.
There is the separate issue whether Phoenix should associate itself with the Reactive Manifesto in the way the Play framework has.
I believe that when Dave Thomas and his co-signers published the Manifesto for Agile Software Development they were sincerely declaring their commitment towards positive change in the software industry. They weren’t primarily motivated by attempting to create the commercial consulting gravy train that later ensued (though of course some of them benefited from it) and ultimately lead to the state where “Agile is Dead, Long Live Agility” (youtube).
My personal impression is that the “Reactive Manifesto” doesn’t come from a genuine realization that “there’s something rotten in the state of Denmark”. It seems to be an attempt to spark something similar to the commercial effect that grew alongside the “agile movement” - but more in the vein of the SOA(-tooling) era.