I cannot speak for the pizza-with-fork thing, but Mayor deBlasio's humor, demonstrated at press conferences and announcements, has proven disarming after 12 years of the stern no nonsense style of his predecessor who seemed uncomfortable with humor. Mayor BdB's self deprecating approach laced with simple humor leaning toward the somewhat silly and obvious is something I think many Dads do indeed relate to as a means of interacting and relating with our kids as they move into junior high and high school years. I think it is a means of establishing a normative relationship with children more commonly used by Dads rather than Moms as the kids enter the rebellious adolescent years. Since I think it's goal is to appear somewhat non-threatening and a way to maintain communications and a less authoritarian stance with kids when it is not necessary to be the stern authoritarian dad laying down the socializing law, it seems an intuitively logical approach for the Mayor to take as he explores various personae which he can express in governing. Whether the corny jokes, off the cuff and shamelessly delivered will last is probably not in doubt, it is woven into the warp and woof of who he is. As for the pizza with silverware thing, it does seem more of an affect that may disappear. I cannot believe that he would go into Smiling Pizza on 9th and 7th and gobble a slice with the ubiquitous flimsy NYC pizzeria knife and fork. Maybe dressed for work as he was at Goodfellas on Staten Island he didn't want to make a mess. But as Errol Louis said on NY1 last night, for a guy who is supposed to be NYC's new populist mayor, "What the hell was that ?"
Michael Howard Saul in the WSJ discusses the cringeworthy Dad style of humor with deBlasio aides and NYC comedians here http://on.wsj.com/KaYneD