How does a pilots license of cluster ballooning differ from other ballooning or unpowered flight?
The license required for operating a manned free balloon would be a "Lighter-than-air, Balloon" license (as opposed to a "regular" license, which is usually thought of as being for "regular" Airplanes - the kind that take off and land using runways, as opposed to lakes).
Cluster balloons are, for all regulatory purposes, just "balloons" - the only difference is rather than having one monolithic envelope filed with hot air like the balloons most of us are used to seeing they have a bunch of smaller envelopes, usually filled with helium.  

An Airman Registry Search shows that there is one "John Ninomiya", with a commercial pilot's license for lighter-than-air balloons, which is consistent with what I'd expect.



A lot of the requirements for a private pilot license in balloons are similar to what you need to do to get a private pilot license in an airplane - for example you still need to learn the same regulatory environment, you'll learn to read the same charts, you'll cover flight planning and weather, etc.

The differences are that you'll be learning balloon systems rather than aircraft systems, and the experience requirements under FAR 61.109 are pretty radically different because of the fact that balloon flying is pretty radically different from flying around in an airplane.

You can peruse Part 61 of the FARs for all the specific details on training requirements for lighter-than-air balloon pilot certificates - Subpart E  (Private Pilots) is probably where you want to look.