There is a Seinfeld episode which contains the following dialogue:
Father-priest: Are you ready my son?
George: Yes faddah.
Father-priest: What did you say?
George: What?
Father-priest: I thought you said faddah.
George: I said faddah, I meant Father. Just a little bit nervous.
My google search produced the song "Hello Muddah, Hello Faddah" (see here), as well as some hints that it is pidgin English. Is the dialogue funny because faddah is typically used by kids?
In this instance, the speaker is mocking the speech patterns of some people from New York and New Jersey. The accent that is mocked is not prestigious; it is associated with a lack of education and some crudeness. Priests are well-educated, so it is humorous that this priest is using a "lower-class" accent.
Using the "d" sound instead of "th" is widespread. "Who dat?" is almost a slogan of New Orleans, home of many Creole speakers. French people speaking English often have difficulty with "th", "d" is how they approximate the sound. It is also a speech characteristic of some Blacks from the southern United States and English people from Yorkshire.
The Alan Sherman song, "Hello Muddah, Hello Faddah" derives a lot of its humor from the contrast of this lower-class accent with the classical music used for the melody.
I always thought it was funny because it's a funny-sounding word on its own, it's not the way George would typically pronounce it (so it's a symptom of his nervousness), and it mirrors the accent of the father-priest. There's a joke in the same episode (maybe the same scene?) where he copies the priest's pronunciation of "vast," which is also pretty funny just on grounds of silliness. So my thought is that it's funny for those reasons, although there could be broader cultural references, as you suggest, that I'm unaware of.
