Let $T: V \rightarrow V$ be a linear operator, let $\beta$ and $\beta'$ be distinct ordered bases in the domain, and let $\hat{\beta}$ and $\hat{\beta'}$ be distinct ordered bases in the codomain.
The idea of similar matrices is that we can represent the same linear transformation in different ways depending on the basis chosen. But in everything I have seen, it is assumed that the basis for the domain and codomain in each coordinate matrix representation are the same, i.e. one is trying to determine the matrix $P$ such that $\left[T\right]_{\beta}=P^{-1}\left[T\right]_{\beta'}P$. In that situation, the $i^{\text{th}}$ column of $P$ is the $i^{\text{th}}$ vector of $\beta$ written in terms of $\beta'$.
Is there a process for getting $P$ if all the bases are distinct? In other words, given the distinct ordered bases, is there a natural way to find $P$ such that $\left[T\right]_{\beta}^{\hat{\beta}}=P^{-1}\left[T\right]_{\beta'}^{\hat{\beta'}}P$?