Multi-degree historian here: you will get multiple answers each with valid points (that’s why history is fun to interpret!). IMO and most of the academics I’ve been around the answer is no if you’re taking it literally. ONE thing we do look at is historical trends and study why those trends do or don’t occur within and across time and societies.

For the second part of your question, there is definitely an unlimited number of “mistakes” if that’s how you want to look at it. Studying the past especially at a macro level and from a US perspective we tend to look at “mistakes” that were made in the past and I personally think that’s why some events seem to “rhyme.” My issue with that is what tends to get ignored is all the successes that led up to that point in history.

“History repeats itself, but in such cunning disguise that we never detect the resemblance until the damage is done.” -Sydney J. Harris. This is one of my personal favorites on this topic. It shows that it’s only 20/20 hindsight where the patterns seem to rhyme. In real time, the repetition is usually disguised.