Nancy, there are many ways to deploy DaaS. Products like Amazon Workspaces, VMware Horizon View, and many others can deliver desktops to the users.
The workstations typically don't change names on each access per se, but are usually in what would be a ""Dynamic"" pool of desktop, and the end user could access any of the desktops within that pool. Again, that is only a function of how the pool is configured. Depending on the needs, some organization also set a ""Persistent"" desktop, which is dedicated for the user, and will maintain its name across sessions.
I would wager that if the pool is set to Dynamic, you're going to be dealing with more issues than just the name of the workstation changing, but rather the software configuration on each. (again, depending on configuration).
To answer your question, this technology has been around for years, more typically deployed within an organization, and is now becoming more available in cloud hosted solutions as well. (i.e: Amazon Workspaces)
Hope this helps.