I encountered this affix for the first time today, in the phrase 半分ごっこ, which was translated as 'sharing'. I suppose that makes sense, if the "something done together" is each having a half of something.
"ごっこ遊び（Gokko-asobi）"(When children play in groups, imitating the work and roles of adults.)
In other words, such as such as "Medical Play," "policeman pretend", when the "imitation" is the subject of the play, in many cases the use of the word.
In the song that was mentioned earlier,
The appearance of "saying goodbye to each other many times" is likened to the appearance of "a child who learns new words and proudly repeats those words."
It is a coined word created by the person in the song, who "can only repeat goodbye" and ridicule and irony, and "play with pretend".
examples shown in japanese dictionary for "something done together" as one of ごっこ meaning are quoted from old novels (mostly before meiji time) so I suspect it's an archaic use of the word, or at least it became extremely rare to be used that way nowadays
In the world of literature and lyrics, "○○ pretend" is often used to mean "like a child's play."
Example: （”戦争ごっこは終わりにしよう” Let's end the war pretend.）
Also, the meaning changes from "Cat and mouse", which indicates barren repetition or infinite loop, and is translated as (いたちごっこ).
About ごっこ
I've been wondering about the affix ごっこ when it's a prefix and when it is a suffix.
There's the word ごっこ遊び, for example, that means "children's game" or "game of make-believe"; but there's also a song named さよならごっこ that would mean, according some translations, something like "make-believe farewell" or "pretending to say goodbye". But the question is: I think that translating さよならごっこ as "pretending to say goodbye" is purely interpretative because of the context of the song; I say that because it's written in Jisho that ごっこ can also mean "something done together". I, however, have not found even one example that ごっこ has this meaning. Can someone tell me some expressions in that ごっこ has the meaning of "something done together"? If there's any rule about the meaning that ごっこ plays in a phrase, please tell me too.