What is the meaning of Spirit ###?
Taking your questions in order:


  Do people who have flown in B-2 Spirits have the Air Force handle "Spirit ###?"


It's plausible. The author of the article says so and there doesn't seem to be any reason to lie about such a trivial detail. That said, he also refers to the pilot as "Scar" rather than as "Spirit [whatever]" so even if Spirit-number is a callsign that a pilot has earned, it doesn't mean it's necessarily the one that they'll always fly with.


  If so, which regulation/directive/etc. specifies that the handle of people who have ever flown in the cockpit of a B-2 Spirit be "Spirit ###?"


There seem to be no such regulations, or if any do exist, they are applied individually. Pilot callsigns are generally given informally, according to this article, which is not an official military publication. Although the content of this article is unfortunate, it substantiates the first one. Finally, the Air Force publishes official voice call sign guidelines, which do not list regulations for pilot callsigns.