Consider $$K=\left\{ \frac 1 n \,\middle|\, n\in\mathbb N\right\}$$
equipped with the standard metric $d_K(x,y) = |x-y|$. Then $(K,d_K)$ is isometric to your space $(\mathbb N, d)$ via $\mathbb N\to K, n\mapsto \frac 1 n$. Open balls in $(K, d_K)$ are easy to visualize, since they are just the open balls of $\mathbb R$ intersected with $K$. This should give you an idea how the open balls in $(\mathbb N, d)$ look.
This is a mistake in the book. As you correctly found, for the metric
$$d^{\ast\ast}(x,y) = \lvert x_1 - y_1\rvert + \lvert x_2 - y_2\rvert$$
the open balls are open squares (with the diagonals parallel to the coordinate axes). This metric induces the standard topology on $\mathbb{R}^2$, hence no nonempty finite set is open. Common names for this metric are $\ell^1$-metric (since it is induced by the $\ell^1$-norm), Manhattan metric (since Manhattan is somewhat famous for a more or less rectangular road network, so the distance one has to travel between two points is the sum of the north-south distance and the east-west distance) or Taxicab metric.
They probably intended to give the metric as
$$d^{\ast\ast}(x,y) = \begin{cases}\qquad 0 &\text{if } x = y \\ \lVert x\rVert + \lVert y\rVert &\text{if } x \neq y \end{cases}$$
where $\lVert\,\cdot\,\rVert$ is a norm on $\mathbb{R}^2$ (often the Euclidean, aka $\ell^2$, norm). This metric has the stated property, for $x \neq 0$ and $0 < r < \lVert x\rVert$ the open ball $B_r(x)$ is the singleton $\{x\}$.
This latter metric is - in my opinion unfittingly - also known as the British Rail metric, SNCF metric, or post office metric. These names are unfitting, because if two points lie on the same line, one doesn't need to travel via London or Paris respectively to reach one's destination, whereas the metric says one does have to.
A more fitting name, under which I first encountered this metric, but which unfortunately doesn't seem to have spread, is the metric of Gaul fishmongers. ("Das Meer? Was hat denn das Meer mit meinen Fischen zu tun?" My translation: "The sea? What has the sea to do with my fish?"; because he buys the fish for his shop in Paris rather than fishing in the sea right behind the village as Asterix suggested.)
Best Answer
Define $r(x) = \min \{d(x,y): y \in X, y \neq x\}$. This is a minimum of finitely many strictly positive numbers (as all $d(x,y) > 0$ when $x \neq y$). So $r(x) > 0$.
Suppose $y \in B(x,r(x))$ and $y \neq x$. Then by definition of being in the ball $d(x,y) < r(x)$ but $r(x) \le d(x,y)$ by definition of $r(x)$. Contradiction. So $B(x, r(x)) = \{x\}$ and the latter set is open.