Having ridden both on a lot of ice for years, I'll answer that most of the fatbikes I have ridden tend to be more stable on ice than a MTB. The wider your tires are (and at lower pressure) the less likely that the wheel will slip (side to side) easily. Riding on snow and ice are very separate things.
Snow is actually very easy to ride provided it is either (semi)packed or not deep. Fatbikes were to some extend perfected by snow riders since the main advantage they provide is not "punching through" packed snow on trails as easily as a narrower tired bike.
Ice can be very difficult to nearly impossible to ride on. Flat ice (like a skating rink) can be ridden across in shorter sections with only good technique. By maintaining balance and not turning or pedaling in a sloppy manner, flat ice can be travelled. Sloped ice (like overflow or an icy hill) requires studs; pretty much end of story. There are many companies out there producing high quality (and some low quality) studded bicycle tires. I have a narrower set (3.8 inches) I use on my fatbike for commuting or riding trails with longer overflow sections.