Let $X/\mathbb{F}_q$ be a variety with an open subvariety $U \subset X$ and a closed complement $Z = X \setminus U$. Then $\#X(k) = \#U(k) + \#Z(k)$ for all finite extensions $k/\mathbb{F}_q$. In particular, if $X$ and $Y$ are varieties that can be stratified into isomorphic locally closed subsets, then $\#X(k) = \#Y(k)$. This kind stratification forgets most of the information of the singularities of $X$.
For example, consider the the nodal cubic $X = \{y^2 - x^2(x - z) = 0\}$ in $\mathbb{P}^2_{x,y,z}$. This is singular at $p = [0,0,1]$ but $X \setminus p \cong \mathbb{G}_m$ and so $\#X(\mathbb{F}_q) = \#\mathbb{G}_m(\mathbb{F}_q) + 1 = \#\mathbb{A}^1(\mathbb{F}_q)$. From the point of view of point counts, $X$ and $\mathbb{A}^1$ are indistinguishable. For a normal example, consider the $A_1$ singularity $X = \{xy + z^2 = 0\}$ in $\mathbb{A}^3$ and let $p = (0,0,0)$, $U = X \setminus p$. Then one can similarly compute that $\#U(\mathbb{F}_q) = (q-1)(q+1) = q^2 - 1$ so $\#X(\mathbb{F}_q) = q^2 = \#\mathbb{A}^2(\mathbb{F}_q)$.
This kind of phenomena is captured by the Grothendieck ring of varieties $K_0(Var_k)$. Its generators as an abelian group are isomorphism classes $[X]$ of varieties over $k$ with the relation that $[X] = [U] + [Z]$ whenever $U \subset X$ is an open with closed complement $Z$ and the ring structure is induced by $[X \times Y] = [X][Y]$. If $k$ is a finite field, then there is a canonical ring homomorphism
$$
K_0(Var_k) \to \mathbb{Z} \quad [X] \mapsto \#X(k).
$$
There is also a homomorphism of abelian groups
$$
(K_0(Var_k),+) \to (1 + \mathbb{Z}[\![t]\!], \times) \quad [X] \mapsto Z^{HW}_X(t)
$$
where $Z^{HW}_X(t)$ is the Hasse-Weil zeta function of $X$ which encodes $\#X(k')$ for all finite extensions $k'/k$. So the point counts over all extensions cannot tell apart varieties with the same class in $K_0(Var_k)$. In the examples above, the class of the nodal cubic equals $[\mathbb{A}^1]$ and the class of the $A_1$ singularity equals $[\mathbb{A}^2]$.
On the other hand, one can produce singularity invariants by point counting or taking the class in the Grothendieck ring of varieties of some moduli spaces associated to the singularity. The approach most closely related to the cohomology of the Milnor fiber is to consider arc spaces of the singularity. This leads to the theories of p-adic and motivic integration and the monodromy conjecture. There is a huge body of literature on this (and I am far from an expert) but the book Motivic Integration by Chambert-Loir, Nicaise and Sebag is a great place to start.
