Why spherical coordinates is not a covering?

No matter how you slice it (mathematicians' or physicists' conventions; latitude or colatitude, etc., etc.), spherical coordinates fail to be a covering map near the "poles", usually the points on the $z$-axis, where every longitude corresponds to a single point. (What time zone does the north or south pole of the earth lie in...?)

In your figure, fixing $r = 1$, you have $$ (x, y, z) = \sigma(\theta, \varphi) = (\cos\theta \sin\varphi, \sin\theta \sin\varphi, \cos\varphi). $$ (I've used "$\sigma$" for "spherical" rather than "$\phi$", in case my fingers mis-type "$\varphi$".)

The "non-covering" happens because if $k$ is an integer, then $\sigma(\theta, k\pi) = \bigl(0, 0, (-1)^{k}\bigr)$ for all real $\theta$. (As a consistency check, it's easy to verify the differential $D\sigma$ has rank one along the lines $\varphi = k\pi$.)

What's true is: For every integer $k$, the restriction $\sigma:\mathbf{R} \times \bigl(k\pi, (k+1)\pi\bigr) \to S^{2} \setminus \{(0, 0, \pm1)\}$ is a universal covering map for the sphere minus the north and south poles, a surface diffeomorphic to a cylinder. Each open rectangle of "width" $2\pi$ (and "height" $\pi$) in this strip is mapped diffeomorphically onto the sphere minus a closed half of a great circle joining the north and south poles.

(I haven't digested your edit carefully enough to know for certain what you mean by $T^{2}/\mathbf{Z}^{2}$, but in the usual reckoning, the integer lattice descends to a single point of the torus.)

Edit: In case a picture helps, the torus below is mapped, via its Gauss map, to the unit sphere. The "north" and "south" latitudes (in blue) are squeezed to points. Their complement wraps around the sphere twice; the shaded "inner" portion maps with degree $-1$, the unshaded "outer" portion maps with degree $+1$.

If $1 < R$, the torus can be parametrized by \begin{align*} \tau(\theta, \varphi) &= \bigl((R + \sin\varphi) \cos\theta, (R + \sin\varphi) \sin\theta, \cos\varphi) \\ &= (R\cos\theta, R\sin\theta, 0) + (\cos\theta \sin\varphi, \sin\theta \sin\varphi, \cos\varphi) \\ &= (R\cos\theta, R\sin\theta, 0) + \sigma(\theta, \varphi). \end{align*} The torus is a realization of $\mathbf{R}^{2}/(2\pi\mathbf{Z})^{2}$, and the parametrization $\tau$ is a universal covering map.

The summand $\sigma(\theta, \varphi)$, which coincides with spherical coordinates, is the value of the "outward-pointing" Gauss map at $\tau(\theta, \varphi)$. The Gauss map itself "discards" the term $(R\cos\theta, R\sin\theta, 0)$, and may be viewed as radially translating the longitudinal sections of the torus so they have a common axis (as if throttling a Slinky), so each blue point maps to a pole of the unit sphere.

Perhaps this makes clearer why the spherical coordinates map fails to be a covering map on any open set containing a point $\varphi = k\pi$ for some integer $k$.

There is nothing wrong with that notation (assuming the mathematical one, according to the Wikipedia link you included). Note that when $\theta=0$, increasing angle $\varphi$ goes from the Z axis to the X axis. In other words, the negative X axis corresponds to $\varphi=\frac{3\pi}{2}$ and, hence, $(-1,0,0)$ does belong to the circle.