Is $A \leqslant_P B \iff A \in \mathsf{P}^B$? If not are there counter-examples?
Assuming $\le_P$ is a polynomial many-one reduction (as opposed to a Turing reduction), the statement is incorrect.
For example, for any language $L$ we have $\overline{L} \in P^L$ (and in fact, in $O(1)$ time and space as well, so this is also an example for logspace many-one reductions)
But obviously if we take $L=\mathrm{Halt}$ to be the halting problem language, we wont have that $\overline{L} \le_P L$ since $\mathrm{Halt}\in \mathsf{RE}$ but $\overline{\mathrm{Halt}}\notin \mathsf{RE}$ and $\mathsf{RE}$ is closed under $\le_P$.