I have several computers with a common home directory on a shared network volume.
Is it safe to run Ubuntu MATE desktop on all these computers at the same time? Will the separate desktop machines conflict with each other when using config files in my home directory?
You can lose changes so how safe it is is up to you, ie if you dont really mind. EG: if you open firefox on a pc and have it configured to remember your tabs, and then open it again on another pc then whichever browser you close last would "win" and next time you open the browser only the tabs from the last pc would be remembered and the other pc history would be lost. Things like that would happen with various programs that save data to the home directory when changes are made or they function. Maybe that doesnt bother you IDK. Im not 100% sure how things like screen resolutions for multi monitor would pan out, as those sort of things are also saved to the home directory so if different pc's have differing multi monitor layouts that could be annoying. I mean probably nothing bad would happen but it sounds fairly annoying to live with.
I suggest each computer having it's own home directory and use your network drive for for shared file storage. I have 2 U-MATE machines, 1 Win 10, and 2 Macs all sharing files storage on my network drive.