A melhor maneira de resolver isso é montar sem uma letra de unidade e atribuir uma letra (menos provável de ser usada por algum processo interno do Azure, como V) com um segundo comando.
Mount-DiskImage -ImagePath "C:\ISO\My.iso" -NoDriveLetter
Get-Partition -DiskNumber 3 | Set-Partition -NewDriveLetter V
Conforme mencionado no meu comentário e no link abaixo, esse é o comportamento esperado, a unidade E: é usada pelo Azure nas imagens do mercado. Se você gostaria de ler mais sobre porque eu estive olhando para este problema:
it was a CD-ROM drive used by Microsoft that was causing this behavior. When dealing with some Azure marketplace images (Generalized), Microsoft will use this drive as a bootstrap point to properly license Windows & configure the OS. When the virtual machine agent enter a ready state and everything is fine, this drive is no longer needed and will be purged eventually, therefore vanishing at some point in the lifecycle of the virtual machine. [...] In my scenario, I was creating a virtual machine scale sets and was starting to assign my first data disk drive letter as F. I used an initialization script I took from the Azure documentation. It worked very well until I needed to re-image instances or perform an OS upgrade in the scale sets. When I did this, the instances were back running but with my data disk with letter E instead of F. It goes without saying that this made my application crash because it was referring to a data location of F:... in its configuration.