When Timmy's children were first shown, it was unknown who their mother was, and this was intended by the creators who gave Tammy and Tommy features of Timmy and both of his love interests, Tootie and Trixie Tang (the latter of whom did not appear in Channel Chasers).
In the canon of A Fairly Odd Movie: Grow Up, Timmy Turner!, however, it is shown that Timmy falls in love with Tootie in adulthood. Likewise, the official Nickelodeon website for the FOP movie contains a quiz which indicates that Tootie would name her first child "Tammy" or "Tommy".
Timmy's female fairy form in the comic strip, Elmerella in the May/June 2004 issue of Nickelodeon Magazine bears a striking resemblance to Tammy, with similar clothes and hairstyle.
The name "Tommy" has also been used by several characters to mistakenly call Timmy. It was also the name of his wished-for older brother. Both Tommy and Tammy have names similar to Timmy to keep the mother's identity further hidden.
Tammy and Tommy are similar to Marty and Mitzie, being a pair of siblings with alliterative names who get Cosmo and Wanda in the future, as a proposed end of the series.
The hat that Tommy wears is known as a "Whoopie Hat" or Whoopie Cap, a style of hat popular among college students in the 1920s, then among young boys from the 1930s to the mid-50s. Whoopie Hats were often fashioned out of a Father's used fedora. Jughead Jones from Archie Comics is the most famous character to wear a Whoopie Hat.