A geometric characterization of these matrices would be the following:
An $(n\times n)$-matrix $A$ is the same as a linear transformation $A:\ {\mathbb R}^n\to{\mathbb R}^n$. Provide ${\mathbb R}^n$ with the standard scalar product.
Claim: The condition ${\rm Col}\,A={\rm Row}\,A$ is equivalent to ${\rm ker}\,A=({\rm im}\, A)^\perp$. In particular any orthogonal projection would have this property.
Proof. The condition ${\rm Col}\,A={\rm Row}\,A$ means that ${\rm im}\,A={\rm im}\,A^t$. Take an $x\in{\ker}\,A$. Then
$$0=\langle A x,y\rangle=\langle x,A^t y\rangle\qquad\forall y\ ;$$
therefore $x$ is orthogonal to ${\rm im}\,A^t={\rm im}\,A$. It follows that ${\rm ker}\,A\subset({\rm im}\,A)^\perp$, whence ${\rm ker}\,A=({\rm im}\, A)^\perp$ by counting dimensions.
Conversely, assume that ${\rm ker}\,A=({\rm im}\, A)^\perp$. Take an $x\in{\rm im}\,A^t$. Then $x=Az$ for a $z\in{\mathbb R}^n$, and we have
$$\langle u,x\rangle=\langle u,A^t z\rangle=\langle Au,z\rangle =0\qquad\forall u\in{\rm ker}\,A\ .$$
It follows that ${\rm im}\,A^t\subset({\rm ker}\,A)^\perp={\rm im}\,A$ and therefore ${\rm im}\,A^t={\rm im}\,A$, as $A$ and $A^t$ have the same rank.
Whether $P$ is invertible or not, one has:
$$\DeclareMathOperator{\rk}{rank}\rk PA\le\rk A$$
since the row-vectors of $PA$ are linear combinations of the rows of $A$.
Similarly, $\rk P^{-1}(PA)\le \rk PA$. However $ P^{-1}(PA)=A$, hence
$$\rk A\le\rk PA\le\rk A,$$
which proves equality.
Best Answer
By the rank theorem it suffices to show $\dim(\ker(\mathbf A))=\dim(\ker(\mathbf{PA}))=\dim(\ker(\mathbf {AP}))$. $$\ker(\mathbf{PA})=\{\mathbf x:\mathbf 0=\mathbf{PA}(\mathbf x)=\mathbf P(\mathbf {Ax})\}=\{\mathbf x:\mathbf{A}(\mathbf x)=\mathbf 0\}=\ker(\mathbf A)$$ $$\therefore \dim(\ker(\mathbf {PA}))=\dim(\ker(\mathbf A))$$ Finally, because $\mathbf P^T$ is invertible and the row and column space of $\mathbf{AP}$ have the same dimension then $$\dim(\ker(\mathbf{AP}))=\dim(\ker(\mathbf{P}^T\mathbf{A}^T))=\dim(\ker(\mathbf A^T))=\dim(\ker(\mathbf A))$$ where $^T$ denotes matrix transposition.