It's true that this is not a distinguishing feature. I don't see that as problematic, it's just how things work. Object-oriented programming, like any paradigm, evolved over time, as people built languages with new ideas, and then spent time figuring out how best to make use of them. And there's no rule saying that nobody is allowed to subsequently adopt and adapt a useful idea in other domains. Fuzzy boundaries are part of the natural order, and that is a good thing.
But, if you go back and read the seminal papers, it's clear that the common thread is encapsulation, every bit as much as referential transparency is the core idea that unites all the seminal work in functional programming. The idea was that programs would be decomposed into modules that were empowered to make their own decisions about what code to execute in response to some instruction. And that this was supposed to liberate the programmer from needing to micro-manage a bunch of state manipulation. Not by eliminating state, as is the ideal in FP, but by delegating the responsibility to manage it.
This is why the concept of Beans bothers me. A Bean is a stateful object that throws its state into your face, and forces you to directly manage it. That sort of approach directly contradicts what OOP was originally supposed to be trying to accomplish. For my part, I am convinced that the bulk of the backlash against OOP is really a response to the consequences of that sort of approach having become orthodox in OOP. Which is deeply ironic, because this is an approach that enshrines the very kinds of practices that the paradigm originally sought to eliminate.