First, note that the category of finite length modules on a noetherian local ring $(A, \mathfrak{m})$ is equivalent to the direct limit of the categories of finitely generated modules on $A/\mathfrak{m}^n$ as $n \rightarrow \infty$ (this just says that any finite-length module is killed by a power of the maximal ideal). In other words, the natural inclusion maps induce an equivalence
$$
\varinjlim_n \mathrm{Coh}(\mathrm{Spec}(A/\mathfrak{m}^n)) \rightarrow \mathscr{A}(A)
$$

Next, we will show the statement in the special case where the scheme is irreducible and $p = 0$. Indeed, let $Y$ be an irreducible noetherian scheme with unique generic point $y$. Then the natural restriction map
$$M^0(Y)/M^1(Y) = \mathrm{Coh}(Y)/M^1(Y) \rightarrow \mathrm{Coh}(\mathrm{Spec}(\mathscr{O}_{Y,y}))= \mathscr{A}(\mathscr{O}_{Y,y})$$
is an equivalence of categories (the last equality is because $\mathscr{O}_{Y,y}$ is artinian and therefore all finitely generated modules are finite length).

Now, by definition $\mathscr{O}_{Y,y} = \varinjlim_{U \ni y} \mathscr{O}_Y(U)$. By "spreading out", this implies that the natural restriction map
$$\varinjlim_{U \ni y} \mathrm{Coh}(U) \rightarrow \mathrm{Coh}(\mathrm{Spec}(\mathscr{O}_{Y,y}))$$
is an equivalence of categories. (This is a special case of the direct limit techniques developed in EGA IV$_3$ §8, but can be proved simply by "chasing denominators").

These maps are compatible (as they are all defined by restriction of modules), so we are reduced to showing that the natural functor
$$
\mathrm{Coh}(Y)/M^1(Y) \rightarrow \varinjlim_{U \ni y} \mathrm{Coh}(U)
$$
is an equivalence of categories. Note that the kernel of the natural map $\mathrm{Coh}(Y) \rightarrow \varinjlim_{U \ni y} \mathrm{Coh}(U)$ consists of those coherent sheaves on $Y$ whose restriction to some non-empty open subset is zero, which is exactly $M^1(Y)$ (as $Y$ is irreducible, so any proper open subset has codimension at least $1$). Thus, the functor is at least faithful.

Now, fix some $U \ni y$, define the closed subset $Z := |Y| - |U|$, and consider the restriction map $\mathrm{Coh}(Y) \rightarrow \mathrm{Coh}(U)$. This is an exact functor with kernel equal to the thick Serre subcategory $M_Z(Y)$ consisting of modules whose support (considered as a topological space) is contained in $Z$. We claim that the induced faithful functor
$$
\mathrm{Coh}(Y)/M_Z(Y) \rightarrow \mathrm{Coh}(U)
$$
is an equivalence.

To see that it is essentially surjective, we use Exercise II.5.15 in Hartshorne, or Tag 01PF in the Stacks project: for any quasi-coherent sheaf $\mathscr{F}$ on $Y$ and any coherent subsheaf $\mathscr{G} \subseteq \mathscr{F}|_U$, there is a coherent subsheaf $\widetilde{\mathscr{G}} \subseteq \mathscr{F}$ such that $\widetilde{\mathscr{G}}|_U = \mathscr{G}$. Now, if $\mathscr{G}$ is any coherent sheaf on $U$, we let $j \colon U \hookrightarrow Y$ be the open immersion, and take $\mathscr{F} = j_* \mathscr{G}$, so $\mathscr{F}|_U = \mathscr{G}$. This shows that any coherent sheaf on $U$ is the restriction of a coherent sheaf on $Y$.

To see that the functor is full, let $\varphi \colon \mathscr{G}_1 \rightarrow \mathscr{G}_2$ be a morphism of sheaves. Take $\widetilde{\mathscr{G}_1} \subseteq j_* \mathscr{G}_1$ as above, so $\widetilde{\mathscr{G}_1}|_U = \mathscr{G}_1$. We have a morphism $j_* \varphi|_{\widetilde{\mathscr{G}_1}} \colon \widetilde{\mathscr{G}_1} \rightarrow j_* \mathscr{G}_2$. Its image is a coherent subsheaf $\mathscr{H} \subseteq j_* \mathscr{G}_2$. Now, we can apply the above lemma to the sheaf $j_* \mathscr{G}_2/\mathscr{H}$ (whose restriction to $U$ is $\mathscr{G}_2/\mathrm{im } \varphi$) to find a coherent subsheaf $\widetilde{\mathscr{G}_2}$ of $j_* \mathscr{G}_2$ containing $\mathscr{K}$ such that $\widetilde{\mathscr{G}_2}|_U = \mathscr{G}_2$. Thus, $j_* \varphi$ restricts to a map $\widetilde{\varphi} \colon \widetilde{\mathscr{G}_1} \rightarrow \widetilde{\mathscr{G}_2}$, which restricts to $\varphi$.

Applying this equivalence, we are reduced to showing that the natural faithful functor
$$
\mathrm{Coh}(Y)/M_1(Y) \rightarrow \varinjlim_{U \ni y} \mathrm{Coh}(Y)/M_{|Y| - |U|}(Y)
$$
is an equivalence. This functor comes from the identity functor on $\mathrm{Coh}(Y)$ by compatibility of the various restriction maps. But $M_1(Y) = \cup_{U \ni y} M_{|Y| - |U|}(Y)$ is the (full) thick Serre subcategory of $\mathrm{Coh}(Y)$ consisting of objects of $\mathrm{Coh}(Y)$ which are contained in $M_{|Y| - |U|}(Y)$ for some $U \ni Y$.

Now, if $\mathcal{A}$ is an arbitrary abelian category and $B_i, i \in I$ are a direct system of thick Serre subcategories, the natural functor
$$
A/(\cup_{i \in I} B_i) \rightarrow \varinjlim_{i \in I} A/B_i
$$
induced by the identity on $A$ is an equivalence. This follows by comparing the universal properties of the left and right sides (alternatively, this is easy to see from the construction of the quotient categories as localizations).

This finishes the proof of the case $p = 0$ for irreducible $Y$.

Now, let $Z \subseteq |X|$ be an irreducible closed subset of codimension $p$. Define $M_Z(X)$ to be the thick Serre subcategory consisting of coherent sheaves on $X$ whose (topological) support is contained in $Z$, and $M_Z^1(X) = M_Z(X) \cap M^{p+1}(X)$ to be the further subcategory consisting of coherent sheaves on $X$ whose (topological) support is a proper closed subset of $Z$. Let $z$ be the unique generic point of $Z$, and consider the local ring $(\mathscr{O}_{X, z}, \mathfrak{m}_z)$.

Restriction defines an exact functor $M_Z(X) \rightarrow \mathscr{A}(\mathscr{O}_{X, z})$ with kernel $M_Z^1(X)$, so we have a faithful exact functor
$$
M_Z(X)/M_Z^1(X) \rightarrow \mathscr{A}(\mathscr{O}_{X, z})
$$

Let $Y_1$ be the closed subscheme of $X$ given by $Z$ with the reduced closed subscheme structure, and $Y_n$ the $n$-th infinitesimal neighborhood of $Y_1$ in $X$. This is the closed subscheme of $X$ cut out by the $n$-th power of the ideal sheaf defining $Y_1$. Each of the $Y_n$'s has unique generic point $z$ and $\mathscr{O}_{Y_n, z} = \mathscr{O}_{X, z}/\mathfrak{m}_z^n$.

By the case $p = 0$ for the irreducible $Y_n$'s, we have an equivalence
$$
\varinjlim_n \mathrm{Coh}(Y_n)/M^1(Y_n) \rightarrow \varinjlim_n \mathrm{Coh}(\mathrm{Spec}(\mathscr{O}_{Y_n, z})) = \mathscr{A}(\mathscr{O}_{X, z})
$$

Thus, the inclusion functors $\mathrm{Coh}(Y_n) \rightarrow M_Z(X)$ induce a functor
$$
\mathscr{A}(\mathscr{O}_{X, z}) \rightarrow M_Z(X)/M_Z^1(X)
$$

Chasing the definitions, we can see easily that these functors are quasi-inverse to each other.

Finally, we have an obvious inclusion $M_Z(X) \hookrightarrow M^p(X)$ for any $Z \subseteq |X|$ of codimension $p$. Since $M_Z(X) \cap M^{p+1}(X) = M_Z^1(X)$, this induces a faithful exact functor
$$
M_Z(X)/M_Z^1(X) \hookrightarrow M^{p}(X)/M^{p+1}(X)
$$
Then, taking direct sums of coherent sheaves defines a functor
$$
\bigoplus_{x \in X^p} \mathscr{A}(\mathscr{O}_{X, x}) = \bigoplus_{\overline{x}\mid x \in X^p} M_{\overline{x}}(X)/M_{\overline{x}}^1(X) \rightarrow M^p(X)/M^{p+1}(X)
$$

We claim that this is the desired quasi-inverse functor to the restriction functor $M^p(X)/M^{p+1}(X) \rightarrow \bigoplus_{x \in X^p} \mathscr{A}(\mathscr{O}_{X,x})$.

To see this, use the fact that if $\mathscr{F} = \oplus_{i=1}^n \mathscr{F}_i$ is a coherent sheaf on $X$ with $\mathrm{supp}(\mathscr{F}_i) \subseteq \overline{x_i}$ for some $x_i \in X^p$, we have $\mathscr{F}_{x_i} = (\mathscr{F}_i)_{x_i}$, as $x_i \not \in \overline{x_j}$ for $i \neq j$.