Are multiple anecdotes still logically unsound?
Honestly, it depends on the context of logic. Anecdotal evidence isn't a cut and dry matter. From the article:
For instance, in medicine, published anecdotal evidence by a trained observer (a doctor) is called a case report, and is subjected to formal peer review.2 Although such evidence is not seen as conclusive, researchers may sometimes regard it as an invitation to more rigorous scientific study of the phenomenon in question.
In science under the logic of induction, the nature of the anecdote would determine its degree of strength. Science is greatly concerned with observations, and in science, a single anecdote might form the backbone of a counterexample or a "major piece of the puzzle". A medical doctor who provides testimony may help confirm or disconfirm a conclusion during differential diagnosis.
For instance, in a court of law, expert witness evidence might include testimony that is anecdotal, and that might be enough to make or break an argument in the eyes of the trier of fact. Again:
Expert witnesses are present in litigation to explain complicated scientific issues, not to influence the jury or judge with fervor. The main responsibilities of expert witnesses are to evaluate potential problems, defects, deficiencies, or errors only when able to fully appreciate a process or system.
And not all use of anecdote is invalid. Many simple minds reason like this. Someone mentioned an authority in their argument. Appeal to authority! He's a manager. Appeal to Force! There's an anecdote in the argument. Argument from anecdote! What makes the anecdote fallacious is it's use as a type of hasty generalization; that is, the argument seems to hinge on the implicit assumption that the anecdote asserts itself as infallibilism. For instance:
Johnny Badlogic: I met a brown bear once, and it was obvious that brown bears are gentle unlike their polar cousins since the bear was kind to me.
Billy Reasonable: You understand that the bear in question was trained for movies and raised his whole life among people, right?
Johnny Badlogic: A bear's a bear's a bear!
Contrast that to this:
Peter the Scientist: I've been studying otters all of my life, and I've only seen an otter in the wild attack a person once. Generally, they flee when people come around. I think the safe conclusion is that otters do not have a disposition (SEP) to attacking people.
Mary Judgesthingsforaliving And your conclusion therefore is that the witness's claim that he was chased down the street by 20 otters necessitating his breaking into the gun shop seems highly improbable.
Peter the Scientist: Well, while attacks by otters do happen on occasion, that claim seems to presume a certain level of determination that has never been reported of otters. And certainly the claim they followed upstairs and trapped him on the roof shortly before the police arrived and then ran off without a trace seems... farfetched?
Lastly, remember that as anecdotes are testimony, multiple anecdotes often form the back ground of scientific research, particularly under the heading of self-reporting. Longitudinal studies conducted to study topics, such as happiness in positive psychology, are one of the backbones of psychological study, for instance. One anecdote in an argument about UFOs is one matter. But 800 people contributing anecdotes over 40 years of their life is another. There's significant strength of epistemic validity in polling particularly when well-devised protocols are observed. Where is the line between a single anecdote and a longitudinal study? Well, that's the sorites paradox at play.
Yes, a million anecdotes don't necessarily prove a truth. Remember, truth is a tricky subject. For instance, self-reporting might be flawed, as the Dunning-Kruger effect demonstrates. And such arguments are likely appeal to popularity:
A majority of the people in the world believe in the supernatural because they have an anecdote. That many people can't be wrong. Therefore, there is a God of some sort.
Consider the same argument with a different topic:
A majority of the people in the world believe the earth is flat based on their lack of experience of it being anything but flat generally. That many people can't be wrong. Therefore, the earth is clearly not an oblate spheroid.
So, a knee-jerk thinker (and we have quite a few of them on this site) will "fallacy monger" when an anecdote enters play and blurt out "Nuh-uh! Anecdote!", but the sophisticated thinkers realizes that anecdotes are testimony, which is a legitimate epistemic source of knowledge, and quality, quantity, and context matter.