If I had to guess I’d say we learned it from them.
However they are still extinct!
It reminds me of the historical narratives in the UK about Viking settlers. We were taught (in the 80s and 90s) to think of the vikings as an invasive force, who were and remained an alien population, who raised levies from the poor, honest britains, and who eventually left or were overcome or just faded from view or whatever. We tended to then skip to the Norman conquest and not talk about it too much. But it's clear in the narrative that the Vikings are 'them' and the saxons are 'us'.
Only when you look at the actual history, the viking people settled and intermarried, cross-pollinated culturally and religiously and are firmly 'us' (if you're British). As a political force, the Norman conquest put an end to their rule of the northern part of England, but it's not like they suddenly all went 'home' after a couple of hundred years of settling.
This is different than the statement that you share 50% of your DNA with your siblings, of course. Because in that case, you actually have the exact identical alleles as your siblings in 50% of your DNA.
The 3% neanderthal DNA is the second type of comparison.
The fact that it has been used without qualification has a lot to do with the fact that most of our genome was assumed to be junk, which we know today is not the case, per the ENCODE project for example.
Thus, the 99% number needs to stop being perpetuated.
Today we know that the alignable parts (parts that are similar enough that they can be aligned with each other) are down in the 80s percents between humans and chimps, which can be digged out from e.g. this recent big study (some numbers are in the paper, and some needs to be digged out of the supplementary material):
Yoo, D. et. al. (2025). Complete sequencing of ape genomes. Nature, 1-18. https://www.nature.com/articles/s41586-025-08816-3
Then remember that a 15%-ish difference with chimps means hundreds of millions of base pairs of difference.
Sometimes hiding complexity behind some ballpark statement can be useful tho. I teach media technology — and a useful simplification is to have students think about inputs and outputs in an abstract fasbion first, then we can talk about signal types and levels and maybe impedances. But in reality a mere piece of wire with a shield can get infinitely more complex and fill a whole academic career. It just isn't useful to start talking about it that way unless you like to get rid of students. I tend to mention simplifications when I use them however, something I wish more scientific journalism did.