What is the name of this type of song?

Cafe Society
1

The song features different characters. The first character sings a line, or couple of lines, or verse. Then the next character sings a different line/lines/verse and the first character repeats the first line/lines/verse. Then character three sings their chunk, followed by character two and character one again.

The examples I’m familar with are comedy/novelty songs. Off the top of my head there’s the novelty song Star Trekkin’. It’s not a round, because the verses don’t overlap and the words of each person’s verse are unique.

2

I think that’s a cumulative song.

3

I’ve heard “The Twelve Days of Christmas” performed in a fashion similar to what you describe.

4

“The Scrotum Song” by the Asylum Street Spankers?

5

Centuries ago, one might sing a motet this way, but no one calls anything a motet anymore.

Several sources imply that the technique the OP describes is now called “counterpoint”, which also had a slightly different meaning in the past.

6

Ah. Misread the OP. Just found a video of “Star Trekkin’”, and that’s not counterpoint, even in the modern sense. If the lines overlapped, that would be counterpoint.

It is more like a cumulative song.