That's not 100% true. There are lots of different theorems that are "central limit theorems", and that work across different cases.
You can have CLT's with non-iid variables (either the aren't identically distributed or aren't independent). The math just becomes much harder, and you have to assume specific dependence structures.
Look for any estimation or inference problems in the context of stochastic processes. For a simpler example you can take a look at sequential test of hypothesis. It is quite ubiquitous, but not always called out by its name.
You can have CLT's with non-iid variables (either the aren't identically distributed or aren't independent). The math just becomes much harder, and you have to assume specific dependence structures.
For example https://en.wikipedia.org/wiki/Martingale_central_limit_theor...