Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3$, with respect to a point not lying on it?

No, you cannot define a winding number if $n\geq 3$ since, as pointed out in a comment by Anthony Carapetis, any two curves in $\mathbb{R}^n\setminus\{ p\}$ are homotopic, and a winding number should be invariant under homotopies. You can however, define a linking number between two continuous disjoint images of spheres in $\mathbb{R}^n$ of dimensions $k$ and $n-k-1$. If $k=1$, the other sphere needs to have dimension $n-2$. In the case of a planar curve, $n=2$ and $k=1$, and the other sphere has dimension $0$ i.e., consists of two points. If one of the points is in the unbounded component of the image of the curve, the corresponding linking number will be equal to the winding number.

It's natural to assume that the winding number is an integer. Nevertheless, this number should behave in a continuous way (see what follows). Consider a curve in $\ \mathbb R^n,\ (n\ge3).\ $ Consider points $p\ q\in\mathbb R^n\ $ outside the curve. Then there exists continuous $ f:[0;1]\rightarrow\mathbb R^n\ $ which is outside the curve, and is connecting $\ p\ q.\ $ The winding number should be constant for points $\ f(t),\ $ hence it should be the same for $\ p\ $ and $\ q.\ $ Thus, under the given understanding, a winding number would be useless.