Azure does run on Windows. It's HyperV server below the rest of the hardware. It's a custom Windows install that you can't purchase, but it's Windows none the less.
I remember seeing an article [1] proclaiming that more than 50% of Azure's workloads are on Linux, so I'm guessing their hypervisors are running an optimised Linux kernel somewhat.
1: Linux is Most Used OS in Microsoft Azure – over 50 percent of VM cores