What's the interpretation of a Poisson regression model coefficient if there is no offset? In a logistic regression model, e.g.,
$$logit(\mu_i) = \beta_0 + \beta_1 X_{1i} + \beta_2X_{2i}$$
where $Y_i \sim Bernoulli(\mu_i)$, $expit(\beta_1)$ is the odds ratio.
In a poisson regression model with an offset, e.g.,
\begin{align*}
log(\mu_i) &= log(\lambda_i t_i)\\
&= log(\lambda_i) + log(t_i)\\
&= \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + log(t_i)
\end{align*}
where $Y_i \sim Poisson(\mu_i)$, $\lambda_i$ is the mean rate per unit time, and $t_i$ is some unit of time, then $exp(\beta_1)$ is the incidence rate ratio.
Now suppose that $t_i = 1$ for all $i = 1, \ldots, n$. So we have
$$log(\mu_i) = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i}$$
My question is, what is $exp(\beta_1)$ called? Is it still the incidence rate ratio (but with a unit time of 1) or is there another word for it?

Yes, it's still a rate ratio (per one unit of time - or whatever else the "per" is, e.g. per 1 km road driving, per 1 m of string etc.).
Of course, this gets harder to interpret, if the time (or whatever else it is) differs between unit of analysis and the offset was just not included. Perhaps, it still makes sense, if the distribution of times is representative for a population (e.g. number of times a human will break a bone in their lifetime over a representative distribution of years lived), but notice that this distribution might change (e.g. people on average live longer nowadays than they used to, but of course lots of other things changed, so I'd not expect to break bones at the same rate as in medieval times).
