Kerr spacetime not symmetric? I always see a term $dt \, d \phi$ in the Kerr-spacetime. Now assuming this means $dt \otimes d \phi$ this means that the Kerr spacetime is NOT(!) symmetric which is somehow non-sense. So do physicists mean that $dt \, d\phi = dt \otimes d \phi + d\phi \otimes dt$ or am I missing anything?

It is standard to denote the symmetric product of two symmetric tensors $\alpha$ and $\beta$ by juxtaposition: $\alpha\beta$. In particular, if $dx$ and $dy$ are coordinate resprentations of 1-forms, then
$$dx\,dy:=\frac{1}{2}(dx\otimes dy+dy\otimes dx).$$
This notation is almost always used implicitly in writing metrics in coordinates. 
Note that with this notation we also have
$$(dx)^2=dx\,dx=\frac{1}{2}(dx\otimes dx+dx\otimes dx)=dx\otimes dx.$$
