Floppy disks formatted in FAT should be visible on any operating system.
I often had problems like this - the dreaded old "The disk is not formatted" message, it usually turned out to be either one of the drives was dying or the floppy drive itself does not support the format (e.g. using 1.4Mb disks in a 720KB drive).
The easiest thing I can recommend is one of the following:
If you have networking on the old machine, try installing an ftp client on the Win3.1 machine and Filezilla on the XP machine.
If it has a network card but it isn't set up, try using a Linux live CD to again FTP the data across.
If both machines have a serial cable, try to track down an old copy of laplink or similar.
There are a few other options, but if worse comes to the worse, the easiest thing you can do is to unplug the hard drive and put it an USB dock or straight in to the new machine and copy across.