The number of grid points near a circle.

Let's look at the 1st quadrant (and then multiply by 4). For each $x$ value from $1$ to $r\sqrt2/2$, there is exactly one $y$-value, and it's greater than $r\sqrt2/2$. For each $y$-value from $1$ to $r \sqrt2/2$, there's exactly one $x$-value, and it's greater than $r\sqrt2/2$. So that gives you $r\sqrt2$ points in the first quadrant, and proves your observation.

Edited due to missing "$r$" in 2nd line.

Here's a slightly different perspective. The $L^\infty$ arc length of the (ordinary) circle of radius $r$ is exactly $(4\sqrt2)r$. So when we define $n$ using a discrete grid, we get the approximation $n=(4\sqrt2)r+O(1)$, and $$\lim_{r\to\infty}\frac{(4\sqrt2)r+O(1)}{r}=4\sqrt2.$$

(This is a suggestive argument, rather than a full proof. I don't know the precise hypotheses we would need to work with more general curves.)