Does classic outlier detection assume normality? My classmate told me he was showing his work in some stuff statistics-based and some time he was showing a boxplot and using it as outlier detection then his professor said 'it's not even correct, the 1.5 from classic outlier formula comes up due to normal data' the classic formula is:
$IC=(Q_1-1.5.IQR, Q_3+1.5.IQR)$ where $IQR=Q_3-Q_1$ and $Q_i$ as $i-th$ quartile.
So all numbers which don't lie in this interval is go as an outlier.
The I get interested where outlier formula come from and why does that assume normality, this image below is as far as I could go in my internet searching:

Finally my question is Has actually outlier classic evaluating come from normal distribution?
, and if so where are the scientific works which shows a proof for that?, or Has there been some incorrect stuff which I've been told on conversation with my classmate?

I won't speak for "classical outlier detection" (methods), because you did not specify them exactly, but as far as the IQR (rule) is concerned, it has no assumptions of normality. In fact one of the main points of this method is that it is robust, since it makes no use of means, which would be an issue in case of extreme values ("outliers").
So to answer your specific question about IQR (rule), no, there is no such assumption. Whether an IQR can be used to detect "outliers" is a very different question, since you may notice that the IQR (rule) is... quite arbitrary.
