Let $X$ be a smooth projective variety, $V\subseteq X$ be any irreducible subvariety and $D\subseteq X$ be a prime Cartier divisor. Assume that $V$ and $D$ meet properly in $X$. Let $Z$ be an irreducible component of the intersection $V\cap D$. Set $A=\mathcal{O}_{X}(U)$, where $U$ is an affine open subset of X meeting $Z$ and such that $D$ is defined in $U$ by a local equation. Denote by $\mathfrak p\subseteq A$ the prime ideal corresponding to $V$ and by $a\in A_\mathfrak q$ the local equation of $D$. Then the intersection multiplicity of $V$ and $D$ along $Z$ is $$\mu_Z(V,D)=\textit{component at } \mathfrak q \textit{ of } (A/\mathfrak p)\cdot (a):=\ell_{A_\mathfrak q}\big(A/(\mathfrak pA_\mathfrak q+ aA_\mathfrak q)\big),$$ where $\mathfrak q$ is the prime corresponding to $Z$. It easy to check that the multiplicity defined above coincides with the usual one (use Koszul complexes). If we take a locally complete intersection irreducible subvariety $W\subseteq X$ in place of $D$, provided that the intersection with $V$ is proper, we can define the intersection multiplicity of $V$ and $W$ at an irreducible component $Z$ as $$V\cdot W=(A/\mathfrak p)\cdot (a_1)\cdots (a_k),$$ where the $a_{i}$'s are the local equations defining $W$ in some affine open subset $U\subseteq X$ meeting $Z$. Also in this case, one can prove that the intersection above coincides with the one defined by means of the Serre-Tor formula.

I sketched a construction that you can find in Thomas Geisser's paper Motivic Cohomology, K-Theory and Topological Cyclic Homology.

Let $X$ be a smooth projective variety, $V\subseteq X$ be any irreducible subvariety and $D\subseteq X$ be a prime Cartier divisor. Assume that $V$ and $D$ meet properly in $X$. Let $Z$ be an irreducible component of the intersection $V\cap D$. Set $A=\mathcal{O}_{X}(U)$, where $U$ is an affine open subset of X meeting $Z$ and such that $D$ is defined in $U$ by a local equation. Denote by $\mathfrak p\subseteq A$ the prime ideal corresponding to $V$ and by $a\in A_\mathfrak q$ the local equation of $D$. Then the intersection multiplicity of $V$ and $D$ along $Z$ is $$\mu_Z(V,D)=\textit{component at } \mathfrak q \textit{ of } (A/\mathfrak p)\cdot (a):=\ell_{A_\mathfrak q}\big(A/(\mathfrak pA_\mathfrak q+ aA_\mathfrak q)\big),$$ where $\mathfrak q$ is the prime corresponding to $Z$. It easy to check that the multiplicity defined above coincides with the usual one (use Koszul complexes). If we take a locally complete intersection irreducible subvariety $W\subseteq X$ in place of $D$, provided that the intersection with $V$ is proper, we can define the intersection multiplicity of $V$ and $W$ at an irreducible component $Z$ as $$\mu_Z(V,W)=(A_\mathfrak q/\mathfrak pA_\mathfrak q)\cdot (a_1)\cdots (a_k),$$ where the $a_{i}$'s are the local equations defining $W$ in some affine open subset $U\subseteq X$ meeting $Z$. Also in this case, one can prove that the intersection above coincides with the one defined by means of the Serre-Tor formula.

I sketched a construction that you can find in Thomas Geisser's paper Motivic Cohomology, K-Theory and Topological Cyclic Homology.