Is it true that Poisson regression is used to model count data? But not all count data follow a Poisson distribution? Then you can still use Poisson regression in that case?
There are alternatives to Poisson regression, which are typically more general. The Poisson distribution, after all, has only a single parameter, and is equidispersed, which is often a very questionable constraint, so most generalizations allow for overdispersion.
One of the more common generalizations is the Negative Binomial distribution, where typically one of the two parameters (e.g., the mean), depends on regressors, usually via a log link, while the other parameter (e.g., the overdispersion parameter) is typically estimated as fixed.
The Negative Binomial distribution is actually an example of a Poisson mixture, i.e., a compound distribution: if the parameter of your Poisson distribution is itself gamma distributed, then the full distribution is negbin. This suggests alternative Poisson mixtures as another way of generalizing the Poisson.
Yet another way to generalize the Poisson is the Conway-Maxwell-Poisson distribution.
Finally, if you suspect that your data originate from different sources, one of which is responsible for only zeros, then you could look at Zero-Inflated Poisson (ZIP) regression. Hurdle models are similar. Again, zero inflation can be combined with other distributions, yielding, e.g., zero-inflated negbin regressions.