Yes it can. In brief, you install Anzio on the file server and then set up shortcuts on the individual PCs which are sharing the file server's disk, to point at the Anzio software.
You can set up individual default files (if you do, reference them on the shortcut command line, e.g., "N:\Program Files (x86)\Anzio17\anzio32.exe" my.def) or let Anzio use its common default file name ("anziowin.def" for both AnzioWin and Anzio Lite), by just referencing the executable.
Or you can have all the instances of Anzio share a common settings file on a network drive (e.g., "N:\Program Files (x86)\Anzio17\anzio32.exe" N:\settings\standard.def). In this last case, you may want to make standard.def a read-only file to prevent one user from changing settings.
More details on Anzio's command line parameters are in the Anzio manual, section 10.2.
Note that AnzioWin and Anzio Lite are licensed on a per-workstation basis.