Path uniqueness in undirected graph
I am assuming that you are interested in simple paths only. Otherwise the answer is trivial.

You can solve your problem in time $O(n)$. Let $s$ be your start vertex and $t$ be your target vertex.
Consider the graph as weighted where every edge has weight $0$.

Find any path $P$ from $s$ to $t$ in $G$. This takes time $O(n)$.
If $P$ does not exist then there is no path from $s$ to $t$.
Otherwise set the weight of each edge in $P$ to $1$ and find a shortest path $P'$ in the modified graph. This takes time $O(n)$ using a straightforward modification of Dijkstra's algorithm.

If $P' = P$ then there is a single path from $s$ to $t$ in $G$.
Otherwise there are at least two distinct paths from $s$ to $t$ in $G$.