The anthropic principle can apply to questions about this universe, too.
For example, consider the question asked by early man: why is the world so hospitable to life, and to us? We have breathable air, food, water, a pleasant temperature, flat ground to stand on, gravity to stop things floating around, etc. It seems incredibly unlikely, unless it was by design. It seems to need an explanation - why do the laws of physics conspire to make life possible?
Then we find out about the rest of the universe. Almost all of it is extremely inhospitable: no air, no food, no liquid water, either burning hot or ultra cold, just dead rocks, ice, and dust floating in space. It turns out that the conditions for life are incredibly unlikely. The laws of physics do not make such conditions inevitable.
So how did we make that mistake? We observed that the environment we lived in was particularly conducive to life. It turned out that that was because we can only survive in places that are particularly conducive; our sample is biased by the fact that we could only sample from that tiny subset of the universe where we can survive. So of course any environment we could observe necessarily was one we could survive in, and we could deduce nothing about how likely or unlikely that was from observing it.
In other words, the Anthropic Principle is another name for sampling bias. Extraordinary coincidences can arise without any physical mechanism to explain them when the phenomenon is such that we can only only observe those cases where the coincidence occurs.
Its application to things like the fundamental constants is speculative and controversial. We at present have no explanation for why the constants are as they are. However, given that a universe with different constants would be hostile to life, one possible explanation would be that all the universes where they're different have no observers, and we only see the values we do because they mean that we can be here to observe them. There may be no explanation in physics for us to find for why they are as they are.
Or of course there may be one, and we just haven't discovered it yet. Since as you say we can't test the hypothesis, it can't ever be a scientific conclusion. It is, rather, possibly a question that science cannot answer. As with Turing's halting problem, some searches would be doomed never to give an answer, but give no way to ever tell whether it's doomed or whether we just haven't searched widely enough yet.
The speculation depends as well on Gell-Mann's 'Totalitarian Principle' of quantum physics - "Everything not forbidden is compulsory". Everything that can happen, must happen. The 'sum over histories' picture posits that every conceivable history happens, and those that are 'inconsistent' in some way interfere with one another and cancel out. But this leaves a lot of space for individually self-consistent alternative histories/universes that evolve independently of one another. But since it posits entities exist that can never be directly observed, we cannot experimentally confirm or refute it - we can only rely on aesthetics like simplicity or symmetry to decide if it serves as a good explanation.
Sampling bias is real. It's application to experimental physics is a real issue. Its speculative application to alternatives that are in principle forever unobservable, like other universes, is a logical possibility, but not one science can prove. But there may be other more metaphysical reasons for believing it.