I believe I miss something simple as I'm not quite good in foundations. Please tell me if I use some notion incorrectly.
As you know $\mathbb R$ is defined uniquely up to isomorphism. The theory of $\mathbb R$ is given by the second-order arithmetic, which is categorical and denoted by $Z_2$. The reals also can be defined in set theory. But taking into account the axiom of determinacy (AD) in fact we have two incompatible set theories: ZFC and ZF+AD. The question of existence of non-measurable sets is answered differently in these theories. And so the theories of reals in two set theories are not equivalent. Does it mean $Z_2$ is incompatible with one of them? If yes, then with which of them, and does it compatible with the other? Does it mean the unique (standard) model of $Z_2$ is also model in only one set theory?