Skipping to the end:

> Moreover, the differences aren't statistically significant IMHO, but you could always debate.

> With those results, I won't say that PostgreSQL on one OS is faster than on the other.

And, to answer my own request, I won't advise my client to migrate to Linux because of PostgreSQL performance. (You could argue that there are more tools for PostgreSQL on Linux than on Windows but that was not the point here)

> For me, PostgreSQL performance on Windows is not better nor worst, it's at par with Linux !

> All the results are available on my github.