The confusion stems from a misunderstanding of the notation $$V \sim
f_V$$ which means both (a) $V$ is a random variable with density $f_V$
and (b) $V$ is created by a PRNG algorithm that reproduces a
generation of a random variable with density $f_V$. Each time a
generation $V_i\sim f_V$ occurs in the algorithm from Casella and
Berger, a new realisation of a random variable with density $f_V$
occurs, which is independent from all previous realisations, hence
different from these previous realisations. (Setting aside discussions
about the pseudo-randomness of these generators and the resulting
approximation in the independence statement.) Equivalently, stating that the $𝑉_𝑖$'s are all identically distributed from the same distribution $𝑓_𝑉$ does not mean that their realisations all are numerically identical.
The starting point of the Metropolis-Hastings algorithm is arbitrary, either fixed $Z_0=0$ for instance or random, for instance $Z_0\sim f_V$ [a notation meaning that $Z_0$ is distributed from $f_V$]. This starting value is always accepted. For $i=1$, one generates $V_1\sim f_V$ [meaning that $V_1$ is distributed from $f_V$, independently and thus different from $Z_0$]
$$Z_1 =\begin{cases}
V_1 & \text{if }U_1\leq \rho_1=\min\left(\frac{f_Y(V_1)}{f_V(V_1)} \frac{f_V(Z_{0})}{f_Y(Z_{0})},1\right) \\
Z_{0} & \text{if }U_1 > \rho_1
\end{cases}$$
and $\rho_1\ne 1$ in general. Hence sometimes $V_1$ is accepted and sometimes not. The same applies to the following steps.
To make a toy illustration on how the algorithm applies, take $f_V$ to be the density of a $\mathcal N(0,1)$ distribution and $f_Y$ to be the density of a $\mathcal N(1,1)$ distribution. A sequence of iid generations from $f_V$ is for instance [by a call to R nrorm]
$$0.45735433,-0.99178415,-1.08312586,-0.85762451,0.92186197,-0.50442298,...$$
[note that they are all different] and a sequence of generations from $\mathcal U$ is for instance [by a call to R nunif]
$$0.441328,0.987837,0.386258,0.316593,0.195910,0.2772669,...$$
[note that they are all different]. Applying the algorithm with starting value $Z_0=0$ means considering
$$\frac{f_Y(V_1)}{f_V(V_1)} \frac{f_V(Z_{0})}{f_Y(Z_{0})}=0.9582509\big/
0.6065307=1.579889>1$$
which implies that $Z_1=V_1=0.45735433$. Then
$$\frac{f_Y(V_2)}{f_V(V_2)} \frac{f_V(Z_{1})}{f_Y(Z_{1})}= 0.2249709 \big/ 0.9582509 = 0.2347724 < U_2=0.987837$$
which implies that $Z_2=Z_1$. The algorithm can be applied step by step to the sequences provided above, which leads to
\begin{align*}
\frac{f_Y(V_3)}{f_V(V_3)} \frac{f_V(Z_{2})}{f_Y(Z_{2})}&=0.2053581 \big/0.9582509 = 0.2143051 < U_3 \qquad Z_3 = Z_1\\
\frac{f_Y(V_4)}{f_V(V_4)} \frac{f_V(Z_{3})}{f_Y(Z_{3})}&=0.2572712 \big/0.9582509 = 0.2684800 < U_4 \qquad Z_4 = Z_1\\
\frac{f_Y(V_5)}{f_V(V_5)} \frac{f_V(Z_{4})}{f_Y(Z_{4})}&=1.5247980 \big/0.9582509 = 1.591230 > 1 \qquad\quad Z_5 = V_5\\
&\qquad\qquad\vdots\\
\end{align*}
producing a sequence as, e.g., below (notice the flat episodes in the graph, which correspond to a sequence of rejections).
$\qquad\qquad\qquad$
As a last remark, the only potentially confusing part in the description of Casella and Berger is the very first sentence where the random variables $Y$ and $V$ are not needed. It could have been clearer to state "Let $f_Y$ and $f_V$ be two densities with common support."
