Consider the following argument:
I have been born on Earth, during a time of relative prosperity. The probability that I was born at this moment, of all moments, is very small. Therefore, this is evidence my existence owes itself to a divine being.
Although the probability of being born at a particular moment is indeed small, it's clearly fallacious because something has to be born at any arbitrary moment in order to make the other births improbable. Each person born could use the same logic to assert they owe their existence to a divine being.
I'm sure there's a term for this, I just can't find it. I've tried searching for "selection fallacy" but only come across selection bias, which is different.