Division of Distributions by Polynomials
Hörmander's result shows that ${\mathcal M}_{\mathcal S}$ is an isomorphic copy of $\mathcal S$. Especially, this means that any linear-topological operation one performs on the side of $\mathcal S$ has its analogue on the side of ${\mathcal M}_{\mathcal S}$. In particular, both are Fréchet spaces. Now, ${\mathcal D}_K=\{u\in \mathcal S\mid \operatorname{supp}u\subseteq K\}$ for a compact $K\subset\Omega$ is a closed subspace of $\mathcal S$. In view of $M_P{\mathcal D}_K = {\mathcal M}_{\mathcal D}\cap {\mathcal D}_K$ (note that $\operatorname{supp}\phi=\operatorname{supp}\left(P\phi\right)$ for $\phi\in{\mathcal S}$, since $P$ is a non-zero polynomial) and $\mathcal D(\Omega)= \varinjlim_K {\mathcal D}_K$, one has that ${\mathcal M}_{\mathcal D} = \varinjlim_K \left( {\mathcal M}_{\mathcal D}\cap {\mathcal D}_K\right)$ is an LF space and the restriction $N_P$ of $M_P$ to $\mathcal D(\Omega)$ is a linear-topological isomorphism between $\mathcal D(\Omega)$ and ${\mathcal M}_{\mathcal D}$. Further, the injection of ${\mathcal M}_{\mathcal D}$ into ${\mathcal D}(\Omega)$ is continuous and also open, the latter by an open mapping theorem due to Pták (see Theorem (4,4) here and Edit 1). Hence, the inductive limit topology of ${\mathcal M}_{\mathcal D}$ agrees with the topology induced by ${\mathcal D}(\Omega)$.
Edit 1. Let $K,\,K'$ be compact subsets of $\Omega$ with $K\subset (K')^\circ$ and $\{\phi_m\}\subset{\mathcal D}_{K'}$ be a sequence with $P\phi_m\to 0$ in ${\mathcal D}_{K'}/{\mathcal D}_K$. According to Pták's theorem, in order to show that the injection ${\mathcal M}_{\mathcal D}\to {\mathcal D}(\Omega)$ is open it suffices to prove that $P\phi_m\to 0$ in $\left({\mathcal M}_D\cap{\mathcal D}_{K'}\right)/\left({\mathcal M}_D\cap{\mathcal D}_K\right)$. This is the same as $\phi_m\to 0$ in ${\mathcal D}_{K'}/{\mathcal D}_K$.
We proceed as follows: As ${\mathcal M}_{\mathcal S}$ is a closed subspace of ${\mathcal S}$, we can regard $({\mathcal M}_{\mathcal D}\cap {\mathcal D}_{K'})/({\mathcal M}_{\mathcal D}\cap {\mathcal D}_K)$ as a closed subspace of ${\mathcal D}_{K'}/{\mathcal D}_K$ (see Edit 2). Under this identification, the convergence $P\phi_m\to 0$ in ${\mathcal D}_{K'}/{\mathcal D}_K$ becomes $P\phi_m\to 0$ in $({\mathcal M}_{\mathcal D}\cap {\mathcal D}_{K'})/({\mathcal M}_{\mathcal D}\cap {\mathcal D}_K)$ (because $P\phi_m\in {\mathcal M}_{\mathcal D}\cap {\mathcal D}_{K'}$). Now using the fact that $M_P^{-1}$ descends to a linear-topologial isomorphism from $({\mathcal M}_{\mathcal D}\cap{\mathcal D}_{K'})/({\mathcal M}_{\mathcal D}\cap{\mathcal D}_K)$ onto ${\mathcal D}_{K'}/{\mathcal D}_K$, we arrive at the conclusion. $\Box$
Edit 2. The identification of $({\mathcal M}_{\mathcal D}\cap {\mathcal D}_{K'})/({\mathcal M}_{\mathcal D}\cap {\mathcal D}_K)$ as a closed subspace of ${\mathcal D}_{K'}/{\mathcal D}_K$ makes use of the short exact sequence
$$
0 \longrightarrow {\mathcal M}_{\mathcal D}\cap {\mathcal D}_K \longrightarrow {\mathcal M}_{\mathcal D}\cap {\mathcal D}_{K'} \longrightarrow (({\mathcal M}_{\mathcal D}\cap {\mathcal D}_{K'}) + {\mathcal D}_K)/{\mathcal D}_K\longrightarrow 0
$$
as well as the fact that $({\mathcal M}_{\mathcal D}\cap {\mathcal D}_{K'}) + {\mathcal D}_K$ is a closed subspace of ${\mathcal D}_{K'}$. To see this, let $\{\phi_m\}$ be a sequence in ${\mathcal D}_{K'}$ and $\{\psi_m\}$ be a sequence in ${\mathcal D}_K$ such that
$$
P\phi_m + \psi_m\to \zeta \enspace \text{in ${\mathcal D}_{K'}$.}
$$
Note that $P\phi_m \to \zeta$ in $C^\infty(K'\setminus K^\circ)$, because the $\psi_m$ vanish to infinite order on $\partial K$. The crux is to see that there exists a $\phi\in{\mathcal D}_{K'}$ such that $P\phi = \zeta$ in $C^\infty(K'\setminus K^\circ)$. To prove this claim (see Edit 3), use Whitney's extension theorem, as in Hörmander's paper, to extend $\zeta|_{K'\setminus K^\circ}$ to a function $\hat\zeta\in{\mathcal M}_{\mathcal D}\cap {\mathcal D}_{K'}$. Then $\phi =M_P^{-1}\hat\zeta$ has the desired properties. Given that, one has
$$
\zeta = P\phi + (\zeta-P\phi)
$$
with $\zeta-P\phi \in {\mathcal D}_K$, and is done. $\Box$
Edit 3. Hörmander in his paper provided estimates which involve both local and global aspects. The global aspects are irrelevant in the present context. Locally, Hörmander obtained pointwise estimates of derivatives of $f$ in terms of derivatives of $Pf$. Especially, one has that, for each $r\in\mathbb N_0$, there is an $s\in \mathbb N_0$ such that $$ \|f\|_{C^r(K'\setminus K^0)} \lesssim \|Pf\|_{C^s(K'\setminus K^0)}, \quad f\in \mathcal D_{K'}. \tag{#} $$ The example below will make this clear.
Now, apparently $\zeta/P\in C^\infty(K'\setminus K)$. Utilizing (#) with $f=\phi_m$ shows that $\zeta/P$ extends smoothly from $K'\setminus K$ to the boundary of $K$. All what remains to be done is to extend $\zeta/P$ further from $K'\setminus K^\circ$ in an arbitrary fashion to a function $\phi\in{\mathcal D}_{K'}$. $\Box$
Example. In 2-D, suppose one wants to estimate $f$, $f_x$, $f_y$ in terms of derivatives of $Pf$, where $P=x^2y\,Q$ and $Q(0)\neq0$. Then it holds $$ \|f\|_{C^1(\overline U)} \lesssim \|Pf\|_{C^4(\overline U)} $$ for any sufficiently small $0$-neighborhood $U$ of $\mathbb R^2$. Indeed, replacing $f$ with $Qf$, one can assume that $P=x^2y$. Then $$ f(0) = \frac12\,(Pf)_{xxy}(0), \enspace f_x(0) = \frac16\,(Pf)_{xxxy}(0), \enspace f_y(0) = \frac14\,(Pf)_{xxyy}(0). $$ As $P$ is less degenerate at other places near $0$, the estimate follows.