"Compatible" arguably means different things by context.
As you say, "iPhone compatible headphones" does not imply "headphones that can run stuff from the app store". This is because headphones never run applications(pedants may argue that the DSP baked in to high end noise cancellation headphones counts; but not really) and nobody expects them to. By contrast "IBM compatible PC" is (when the vendor wasn't lying) precisely a statement about a device's ability to run a particular class of software; because the context, that of computers, implies that that is what "compatible" means.
Given that the use of phones to run applications enjoys a higher profile than the use of phones as remote controls(and, even in those cases where they are used as remotes, this is generally done through an application), it isn't wildly unreasonable to assume that "compatible" means "software compatible, at least in some sense".
Given history, I suspect that the headline could be more accurately rewritten to say "Sony-Ericcson releases high-end dumbphone defined by a couple of genuinely interesting features and a lot of mediocrity(just like all the other times they've done that), also functions as a bluetooth remote for one specific home theatre device".