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I think I can explain what's going on by interpreting things more formally, though it could be gotten at informally as well; it's just clearer to see formally.

One way to interpret "$y = \pm x$" (and other informal notations like $y = 1,2,3$) is as $y \in \{x, -x\}$ ($y \in \{1,2,3\}$). This is logically equivalent to $y = x \lor y = -x$. So that's one direct answer.

However, because $(A \lor B) \implies C$ is equivalent to $(A \implies C) \land (B \implies C)$ (and $\forall x \in S. P(x)$ is equivalent to $\forall x. x \in S \implies P(x)$), the text might have a more conjunctive rendition. To be clear, we'd still interpret "$y = \pm x$" as $y = x \lor y = -x$, but a statement like "$y^2 = 1$ for $y = \pm 1$" could mean $\forall y. y = \pm 1 \implies y^2 = 1$ and could be rendered as "$y^2 = 1$ for $y = 1$ and $y = -1$".

Of course, for roots of polynomials we often take something like "$y^2 = 1$ for $y = \pm 1$" to mean $y^2 = 1 \iff y\in\{1, -1\}$. In this case, if we take the left-to-right implication, i.e. using the quadratic equation to constrain the values of $y$, then clearly viewing $y=\pm 1$ as "$y = 1$ or $y = -1$" is natural. Going the other way, i.e. showing that these values of $y$ satisfy the quadratic equation leads to the situation of the previous paragraph.

I don't think I've ever seen anyone write "$y = \pm x$" to mean $y = x \land y = -x$. In most contexts, this immediately implies $y=0$ so you may as well write that. If the point is to prove $y=0$ by showing $y = x \land y = -x$, it would be clearer to write something like $x = y = -x$.