Yes, you can fit any time series (with or without external variables) using HMM, but there are some constraints:
- It should follow the Markov property.
- There is some variance that other models are not able to capture (in other words, the system is partially observable).
Adding to point 1, for HMM, it should hold true, but the way Baum Welch algorithm works, indirectly it considers the values of more than the previous state for HMM (order-1). The state $t-1$ depends on $t-2$, which in turn depends on $t-3$. The calculation of parameters (transition, emission, starting probabilities) happens over multiple iterations and it finds parameters in such a way that holds Markov property true.
I think that, when they say 'any', they mean even when you don't have all variables needed to forecast future values.