What is it that makes bye-bye so inherently juvenile compared to goodbye, or simply bye?
I have to imagine it’s the repetition, isn’t it? Surely it reminds us of babies, repeating simple sounds. Think of how we stereotype baby speech: goo goo, ga ga. Bye-bye.
It’s also probably part of the general trend in English of avoiding statements that are too abrupt. The monosyllabic bye can seem a bit rude. Think about how we usually actually say it: byyeee! We always elongate it to sound less rude (unless we want to show someone we’re annoyed).
Imagine telling a young child to say bye to someone, rather than bye-bye. Weird, eh?