You may want to have a look of the following paper by Kreck and Lueck:

Topological Rigidity for Non-aspherical Manifolds

where they showed that a necessary condition for $S^d\times S^d$ ($d>2$) being Borel (which means $Aut(S^d \times S^d)$ acts on $S^{Top}(S^d \times S^d)$ transitively,or equivalently,there is no "fake" $S^d\times S^d$ in your sense) is that $d$ is odd or $2d+2=2^l$ for some $l$.

Now for $S^{4k+2}\times S^{4k+2}$ ($k>0$),we know the necessary condition above is not satisfied,hence,there is some fake $S^{4k+2}\times S^{4k+2}$.

You may want to have a look of the following paper by Kreck and Lueck:

Topological Rigidity for Non-aspherical Manifolds

where they showed that a necessary condition for $S^d\times S^d$ ($d>2$) being Borel (which means $Aut(S^d \times S^d)$ acts on $S^{Top}(S^d \times S^d)$ transitively,or equivalently,there is no "fake" $S^d\times S^d$ in your sense) is that $d$ is odd or $2d+2=2^l$ for some $l$.

Now for $S^{4k+2}\times S^{4k+2}$ ($k>0$),we know the necessary condition above is not satisfied,hence,there is some fake $S^{4k+2}\times S^{4k+2}$.

For $S^2\times S^2$,it is indeed Borel,i.e.Every manifold which is homotopy equivalent to $S^2\times S^2$ is actually homeomorphic to it.