I don't believe it's unfalsifiable, its just fundamentally true. You cant observe yourself in any reality where you are incapable of onserving yourself.

As you note, it's the only logically-consistent framework through which to view the world we live in. But are there other ways in which the universe could've formed that we might be able to falsify it?

Imagine two universes, A and B. They're entirely disconnected and independent from one another; no matter or energy can flow in either direction, except that through some exotic process, a small window exists in universe A through which universe B can be observed without affecting it in any way (Heisenbergs HATE this!). Universe A is just as our own is, including our existence, with the single exception of this window. In universe B, however, the laws of physics do not permit carbon atoms to form in stars, so no sentient life has ever formed.

In those universes, then, the Anthropic Principle would be falsified; as the residents of Universe A could observe a universe in which they could not have arisen.

Or consider a Boltzmann Brain (or a simulated universe). Were we to discover that our existence was of either nature, that too would falsify the Anthropic Principle, as we are not actually observing a universe.

Anyway. It's not falsifiable in our reality, as far as we can tell. But we can imagine ways in which it could be falsifiable.