If that is happening, you are rebooting into the installer on the ISO file and not to the hard drive.
After installing Ubuntu server to the hard drive, shut down the VM. Then open the VM’s settings in the Fusion GUI, and open the settings for the CD/DVD device. Uncheck the box to connect the CD to the VM.
Now reboot the VM. You should now be running from the hard drive. Now install the Ubuntu-desktop package. Any work you do now will available between restarts.