I think this is supposed to be an example of one ray (that contains all points on the right side of the horizontal axis and the origin) which cannot have a unique pair (𝑟,𝜃) assigned to any of its points.
Yes, I think this is what he means. I was confused by this paragraph as well.
He doesn't really say it, but of course our choice of $\theta$ is somewhat arbitrary, so long as it covers the full 360°-$2\pi$ radians.
We can use $\theta$'s defined by
$$0 \leq \theta \leq 360$$
or
$$73 \leq \theta \leq 433$$
or
$$-\frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}$$
(Analogously, we could choose any ray through the origin against which we measure $\theta$. The choice to use the positive $x$-axis is somewhat arbitrary.)
In general we can use
$$\theta_0 \leq \theta \leq\theta_0 + 360$$
where $\theta_0$ is any initial angle.
I think what he's getting at with his "exclude some ray through the origin"
is that we exclude all points along the ray defined by $(r,\theta_0)$ where $r \geq 0$.
For any choice of $\theta_0$, we exclude the points $(r,\theta_0)$, along with the origin, leaving us only with points that can be unambiguously assigned to unique $(r, \theta)$, where $\theta_0 < \theta < \theta_0 +360$ and $r > 0$.
Analogously, We can chose to measure our $\theta$ against the positive $y$-axis, or the ray that's 76° counterclockwise from the negative $x$-axis, or any other ray through the origin, but regardless of which one we chose we must exclude points along this ray including the origin if we want to be able to assign unique $(r, \theta)$ coordinates.
This of course is not really satisfactory and will leave us with points we can't describe regardless of our choice of $\theta_0$
We could instead, restrict our $\theta$ to either
$$\theta_0 < \theta \leq \theta_0 + 360$$
or
$$\theta_0 \leq \theta < \theta_0 + 360$$
and define the origin point as having some fixed $\theta$, for example $\theta_0$ or $\theta_0 + 360$, depending on our choice of allowed $\theta$. We can easily avoid the ambiguity Spivak's worried about, without excluding any rays through the origin.
Fortunately, it's not really important. He goes on to say that, despite any difficulties in picking points on the plane and then assigning to them unique $(r, \theta)$ coordinates, going the other way is no problem at all.
Given any $(r, \theta)$, this maps to a unique $(x,y)$ in the plane, and the rest of the chapter involves this process.
Best Answer
The confusion is not in what absolute values are, but in what variables are.
The statement $$ -b \le a \le b \iff |a| \le b $$ (with an implied "for all $a,b \in \mathbb R$ at the beginning) means that if we have any values of $a$ and $b$ for which one of the two sides holds, we can deduce that the other side also holds for them. For example, we could take $a =-3$ and $b=5$, and check the right-hand side: $|{-3}| \le 5$. Then we could immediately conclude the left-hand side: that $-5 \le -3 \le 5$.
What we can also do is, for any value of $a$, pick the value $b = |a|$. There is nothing special about absolute value here: we could have picked $b = a^2$, or $b = e^{\sqrt{\log a}}$, or any other expression in $a$. But picking $b = |a|$ is convenient, because then the right-hand side $|a| \le b$ simplifies to $|a| \le |a|$, which we know holds for any $a$.
Therefore the left-hand side, which simplifies to $-|a| \le a \le |a|$, must also hold for all $a$.