I'll try to answer with a few general remarks on what a pole is and what a meromorphic function is.
Let $U \subseteq \mathbb{C}$ be non-empty and open and let $X$ be a complex Banach space (for instance, the space of bounded linear operators in $\mathcal{H}$).
Definition 0. (Pole) Let $V \subseteq U$ be non-empty and open, and let $f: V \to X$ be holomorphic. We say that a point $p \in U \setminus V$ is a pole of $f$ if $p$ is isolated in $U \setminus V$ and if the principal part of the Laurent expansion of $f$ about $p$ is a finite sum.
Note that this definition assumes explicitly that $p$ is isolated in $U \setminus V$.
Definition 1. (Meromorphic function) A meromorphic function on $U$ is a holomorphic function $f: V \to X$, where $V$ is an open subset of $U$ and each point in $U \setminus V$ is a pole of $f$.
Note that, in this case, $U \setminus V$ is automatically discrete since every point in $U \setminus V$ is a pole and thus, by the definition of the notion pole, isolated in $U \setminus V$. So one does not need to assume the discreteness $U \setminus V$ explicitly here, since it's implicitely contained in the assumption that all points in $U \setminus V$ are poles.
However, in his book Borthwick apparently takes a somewhat different perspective on meromorphic functions. I'll rephrase his definition, using what I would consider a slightly more precise wording:
Definition 2. (Meromorphic functions, again) Let us call a function $f: U \to X$ B-meromorphic if the following condition is satisfied: for each point $a \in U$ there exists an open ball $D \subseteq U$ with center $a$, an integer $m \ge 0$ and vectors $A_{-m}, A_{-m+1}, \dots, A_0, A_1, A_2, \dots \in X$ such that
$$
(*) \quad
f(z) = \sum_{k=-m}^\infty A_k(z-a)^k
\qquad
\text{for all } z \in D \setminus \{a\},
$$
where the series converges in norm.
Now, if $f: V \to X$ is meromorphic, we can clearly extend it to a B-meromorphic function by assigning arbitrary values to it on $U \setminus V$. But we also have the converse implication; more precisely:
Proposition 3. Let $f: U \to X$ be B-meromorphic and let $E \subseteq U$ denote the set of all points $a \in U$ at which the number $m$ from the Laurent expansion $(*)$ cannot be chosen as $0$. Then $f|_{U \setminus E}: U \setminus E \to X$ is meromorphic (in particular, $E$ is discrete in $U$).
Proof. It is a consequence of $(*)$ and of the definition of $E$ that, for each $z \in U \setminus E$, the function $f$ is holomorphic in a neighbourhood of $z$ (including $z$ itself); in particular, an entire neighbourhood of $z$ is contained in $U \setminus E$. Thus, $U \setminus E$ is open and $f|_{U \setminus E}$ is holomorphic.
Let us show next that each point in $E$ is isolated in $E$; so assume to the contrary that $E$ accumulates at a point $a \in E$. We know that $(*)$ holds for all $z \in D \setminus \{a\}$, where $D \subseteq U$ is an appropriately chosen open disk with center $a$. In particular this implies that $f$ is holomorphic on $D \setminus \{a\}$.
However, since $E$ accumulates at $a$, there exists a point $a_1 \in E$ which is in $D \setminus \{a\}$. Since $f$ is, as noted in the previous paragraph, holomorphic at $a_1$, the principal part of the Laurent series expansion of $f$ about $a_1$ vanishes. However, this contradicts the fact that $a_1 \in E$.
So we showed that every point in $E$ is isolated in $E$. Hence, it follows from $(*)$ that every point of $E$ is a pole of $f|_{U \setminus E}$. Thus, $f|_{U \setminus E}$ is meromorphic. $\square$
Note that the observations above show that Definition 3 (i.e., the definition from Borthwick's book) has a somewhat strange consequence: the function $f$ is assumed to be defined everywhere on $U$, although it is holomorphic only on $U \setminus E$; in particular, the values of $f$ on $E$ are completely unrelated to the other values of $f$.
