Back when I was still in school there was a room full of Linux machines. The weird thing was that in those machines weren't any hard drives. What kind of sorcery is this? Is there a Linux server with multiple users set up and their display is set to 1 machine? Another weird thing was changing the password. After changing it the machine wouldn't boot, you had to ssh in and change it back to the original one. So how is such a setup called or what are keywords to look for?
These would be diskless workstations or thin clients.
A typical Linux implementation would be Linux Terminal Server Project (LTSP) where all the applications run on the server and the clients just handle the display.
An alternative is using full-power PCs, but having them boot and run from a network hard drive. The operating system and applications all run on the client.
There are a few ways of doing this. Why you couldn't change your password is beyond any of us, it would be down to how the local setup was done.
The lab I teach in we have the Mint 19.x, Ubuntu 16.04, and the Debian netinstall disk set up as being available via PXE boot from a TFTP server. Booting any of these is just like booting the equivalent CD/DVD from a disk or flash drive.
You could also set up installs that mount various shared directories from a NFS server or similar. Possible to pull in home directories, the contents of
/usr, etc. In this case, you have a split file system, some local and possibly read/write, some remote and read-only, some remote and read-write.
It is also possible to boot locally with just enough Linux and a generic user to launch an X server and then query a remote machine via XDMCP, possibly over SSH. This would be used to create "dumb terminals" where all of the GUI programs are running on the remote server with only the local keyboard, mouse, and display in use.
Historically there are lots of other ways like hooking up a bunch of serial terminals, etc. but the chance of this being used depends on how long ago you were in school (I did Cobol and Fortran coding in high school connected via 14.4k modem/serial connection to a VAX at the local univeristy in the early 80s... pre-Linux but same concept).