Let's say I want to model the arrivals of some quantity of interest, say customers coming to a store. I know that on average, $\mu$ customers arrive in on hour. My understanding is that if $N$ is the number of customers to arrive in an hour, then
$$ N \sim \text{Pois}(\mu) $$
Now I think of $\mu$ as a rate, e.g. $\mu$ customers per hour. But I can also reason about the number of customers in $t$ hours, which is
$$ N_t \sim \text{Pois}(\mu t) $$
My confusion is: in the first example, the dimensions of $\mu$ are
$$ \mu\;\frac{\text{people}}{\text{hours}} $$
whereas the dimensions of $\mu t$ are
$$ \mu\;\frac{\text{people}}{\text{hours}} \times t\;\text{hours} = \mu t \; \text{people} $$
What's going on? Is the parameter an average rate or an average count? Or is the first object a Poisson r.v. while the second object is a Poisson process?
The interpretation of any given parameter depends on context. But you can resolve your specific question by viewing $\mu$ as "people per hour" in both cases, and viewing $N$ as the special case $t=1$; that is the "$\mu$" in $\text{Pois}(\mu)$ is actually $\text{Pois}(\mu \cdot 1)$.
In general, the parameter of a Poisson process is a rate (arrivals per unit of time), and the parameter of a Poisson random variable is a counting unit (number of people, number of arrivals, etc.). In your example $\mu$ is a rate, and $\mu \cdot 1$ is a counting unit.