Let's say I install Wubi to a USB. Let's also say that I, while using my USB version of Ubuntu, saved a file.
If I ran that same USB of Wubi on a different PC, will I lose that file?
The answer is yes and no. It depends. If you, for example, create three partitions on your flash USB drive like this:
In such case, if you set each time on every computer installation the same way, your home partition will be untouched.
However, I would recommend not to use Wubi, but an ordinary Ubuntu (or Xubuntu, or even Damn Small Linux) installation on the pendrive. I did the latter once and that worked really well. I've had three computers with Microsoft Windows on each hard drive. On micro-SD cards I had installed 3 Linux systems: Debian Lenny, and on one 8GB flash pendrive I've had my home partition with all my data projects and stuff. When I needed Linux, I just plugged in the micro-SD card and my home partition pendrive before booting, and I would have had a fully functioning Linux distribution. I am sure you could do the same with Wubi.
I think you're confusing Wubi. That installs Ubuntu within Windows. You can install it to a USB but it's booted via the Windows installation (the Windows boot manager). You can't plug that USB in to a new computer and expect to boot it (even if that computer is running Windows), and you cannot boot it as a live USB.
Any file you save on the Wubi install (wherever it's installed) will be permanently saved.
What you're probably thinking of is creating a bootable USB with persistence. This allows you to boot from a USB on any computer, and also any file you save will be available on other computers. This is very different to Wubi. For more information about Wubi please refer to the Wubi Guide.