What's the verb that means: to make an anecdotal statement (based on casual observations) as if it were a fact?
I don't think it's "to lie" because this person believes it's the truth. He is unknowingly stating a fallacy.
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Unknowingly stating a fallacy might be to err, to make a mistake, to be mistaken, or to be wrong. None of those necessarily tie to the "mak[ing] an anecdotal statement...as if it were fact" part of your question, but they're good, general words which don't impute intent the way to lie does.
Treating an anecdote as fact might be overgeneralizing.
You could, perhaps, misspeak, but that has more an idea of "accidentally stating incorrectly or inappropriately", like if you use the name of the person you are addressing in place of the person you're supposed to be speaking about.
Either misstate or misreport, on the other hand, appears to be more appropriate.