I think this scenario is incoherent after thinking about it for a minute. You say "there is equal probability that either explanation is true" but you imply that you, the storyteller, know something about license plates. Probability is a description of incomplete knowledge, so whose knowledge are we talking about, yours or mine?
I see no reason (in my hypothetical ignorance) to exclude the possibility you know I will see "ARW 357" next even if all license plates are different.
Instead, consider the example with the urns. I think that's a lot more realistic and illustrates the problem.
In both cases you do not have enough information to make any conclusions.
That's why the field of statistics exists in the first place.
However, are we able to reduce down our understanding of "which universe we're in" in such simple terms? I think I'd understand better if I knew how long we've been capable of detecting oumuamuas.
There's no evidence it is aliens. Obviously it could be, and it's unlikely there are not such things in our universe.
It just seems more appropriate to say "there's no evidence it is related to alien tech." Everyone having to say "it's NOT aliens" seems like some sort of cultural phenomenon.
Most theories on physics are evaluated this way. "We theorize that light is a particle, which would explain these properties of light, and means we should expect to see these other properties of light if we test them." If further observation of light matches those properties you predicted then that's pretty good evidence that your theory is on the right track.
If your ancient aliens theory didn't actually predict "Stonehenge", but predicted that the aliens would have left "all sorts of cool weird stuff", and then you point at Stonehenge and say "see, that's the sort of cool stuff I mean", then I'm no longer impressed by the predictive power of your ancient aliens theory.
Predictions have to be precise for a theory to be supported by them: they can't be vague generalities. cf. astrologers, who "predict" the future by predicting nothing.
My theory is wrong, but it assigns higher likelihood to Stonehenge than yours. That’s why I don’t like the license plate example.
A theory that is wrong will sometimes predict correct things by accident. That's unavoidable, but we a) have statistical methods for modelling how often things go right by chance, and b) have a philosophy of science in which theories are forever tentatively held: your aliens theory might beat any theory which says Stonehenge is impossible, but will soon be replaced by a theory which explains Stonehenge more economically.
Does this single observation tell you anything about reality? I would argue that it does - it should strongly suggest to you that you're in the "every car is ARW 357" reality. Similarly, if you draw a ball from an urn and it's red, and you know you've drawn either from the "urn of red balls", or the "urn of one million green balls and one red one" you should be pretty confident you've drawn from the "urn of red balls".
If one theory can explain your observations but the explanation is highly unlikely, and another theory explains your observations and the explanation is likely, the second theory is more likely to be correct all else equal.