You're right the two tags are related — as indeed the system shows.
Conversation is an unscripted interaction between two or more persons.
Dialogue, as "a feature of a book, play or film," is a scripted interaction designed to look like a conversation.
The two tags are not synonyms: the scripted nature of one which is absent in the other is an important difference. Conversation is off-the-cuff, unconsciously spoken, while dialogue has had a third party consider it carefully in order that it appears to be instinctive. In reality, it's anything but.
Thus the dialogue of spoken conversation in JK Rowling's Harry Potter books, for example (about which we have had a number of questions) is not the same as conversation which real children have in the schoolyard1.
Nor is there a hypernym available. The tagging system doesn't handle hierarchies; it's only able to make equivalences at the same level. If we invented a hypernym like spoken-interaction and then linked both conversation and dialogue to it, all three would become synonyms. The synonym system is designed to help with common misspellings like dialog for dialogue, or with alternative terms like spelling and orthography.
1 Sorry, Harry Potter isn't real. Perhaps that should have been in spoiler markup.