Actually after writing my other post, I've come across several romance books that did dialogue similar to this. There was still he said she said peppered throughout but there were large portions of the book that didn't have it. So it's not only a Japanese style of writing.That's more of a Japanese style of writing dialogue than anything. Authors usually also have the habit of putting specific styles to each character's way of speaking so as to make it easier for readers to discern who is the speaker. (Like, one character uses desune, another desuyo, another da, another da ze... And so on.)
... And well, since we don't have such easy to discern speech markers in English, there are often times in dialogues with 3+ characters that the speaker becomes confusing in fantranslations that don't add a "he/she said."
Translators sometimes stick to this style in order to give a similar reading experience, or they localize it by putting the usual speech denominators you expect in English novels.
Official translations tend to localize the speech to feel natural to English speakers, mind you, so... The dialogue you see is only present in fantranslations and in the original Japanese, really.