Yes, we have been running IDS in the AWS cloud for almost 10 years. Amazon themselves host IDS in AWS (the Createspace cover creator).
there is not much difference between server and desktop in. The way they open a file may be slightly different. How come Id server cost so much?
id server plugin is model only plugins. While desktop normally had UI such as dialog, panel, etc. components.
InDesign Server has the most powerful composition engine in the world, completely exposed to automation, so while I wish it were cheaper I understand Adobe's perspective and it makes sense for many organizations. Of course there are small businesses for whom it just doesn't make sense.
The price depends on how you use it. The limited license - for batch composition - is less expensive than the premium license. Yet it is more expensive than desktop because it lets you spit out huge amounts of content in an automated way. Running as a server lets InDesign create exponentially more output.
The premium license required for web to print applications ended up more costly because Adobe imagined a very sophisticated web to print app might cannibalize InDesign. We actually see something like that with our Silicon Designer product, yet typically the desire is for a much simpler UI and I doubt we have made a dent in InDesign desktop sales.
While IDS is not inexpensive, the multi-instance license is sort of a loophole given the trajectory of servers. We see clients with 64 cores running throughput on a single multi-instance license that would have required multiple licenses years ago.