Tama, or Tamaya, is named after a famous guild of pyrotechnicians. During the Edo period, they competed with the Kagiya guild to produce the greatest fireworks for the annual summer fireworks festival, held over the Sumida River in modern Tokyo. The festival was originally started in the 1700s to soothe the spirits of the dead who had been lost to recent plague and famine.
The Tamaya guild came to be held in the highest regard, until a fire broke out, and they fell out of favor. But onlookers would still shout the names of their favorite guilds, and this practice continues to this day.
The tradition of guild rivalry in the festival also continues. Fireworks are sometimes even engineered to create patterns such as kanji or even characters such as Pikachu. We see Tama do this as well with the likenesses of Ammy, Issun, Mushi’s mama, and various other characters animals.
EDIT: I just realized that the cloth hanging from his belt actually reads “Tamaya”, with the same characters used by the guild in real life.
EDIT 2: His tattoo (粋) can be read as “iki”, meaning “chic” or “stylish”, or as “sui”, meaning “the best.” Someone thinks highly of himself…
Sources: x x x