If I have a simple graph $G$, and what to count the number of simple paths between two distinct vertices, can the paths be seen as independent sets of a matroid?
Perhaps yes, because the edge space of the graph and the cycles form a matroid (the cycle matroid), but the union of two paths (overlapping or otherwise) forms a series of loops, at least 1. Also, there is the notion of the paths spanning the union of paths, if we think of this as a vector space over $\text{GF}(2)$ (so paths as elements of a basis are either scaled by 0 or 1, i.e. exist of don't exist, and addition means the two paths become the respective union, and multiplication is set up properly i.e. considering the edges as part of vector space).
I do not think this matroid is the path matroid or gammoid. But, crucially, though you may be able to construct a matroid like this, is there a theory of enumerating the independent sets, for example, when the graph is random, in order to get some probabilities for path counts in e.g. continuum percolation?
The set of all "partial simple paths" between two vertices is a simplicial complex. More precisely, if $v,w$ are vertices of the graph $G$ then the set of all simple paths between $v$ and $w$ is a subgraph $G(v,w)$ that can be directed so that $v$ is the unique source and $w$ the unique sink. This directed graph is acyclic so it is the Hasse diagram of a partially-ordered set $P = P(G;v,w)$. Now take the chain complex $\Delta = \Delta(G;v,w)$ of this poset.
The independent sets of a matroid form a pure simplicial complex, that is, a simplicial complex all of whose facets have the same cardiniality. Since $P$ is a bounded poset, $\Delta$ is pure if and only if $P=P(G;v,w)$ is graded if and only if every simple path in $G$ from $v$ to $w$ has the same length. It is easy to construct graphs and select vertices such that $\Delta$ is not pure and hence not a matroid. For example let $G$ be any odd cycle and $v,w$ be arbitrary distinct vertices.
For an example of a triple $D = (G,v,w)$ such that $\Delta(D)$ is pure but still not a matroid take $G = T_{m,n}$ to be the tadpole graph with $m$ even and $m \ge 4$ and take $v,w$ to be the two vertices at distance $m/2+n$. Then $\Delta(D)$ is pure because its two facets both have size $m/2+n$ but it is not a matroid because these facets don't satisfy the matroid basis exchange axiom.
EDIT: The "partial simple paths" in the graph $G(v,w)$ are the chains in the poset $P = P(G;v,w)$. The chain complex of $P$ is also known as the order complex which is a main topic of Wachs' wonderful lecture notes. See page six of those notes for the definition for a full accounting. In short, a "partial simple path" turns out to be a collection of edges in $G(v,w)$ that can be completed to a (directed) path from $v$ to $w$.