There could be a few reasons for this. I believe it mostly comes down to the choice of prepositions. Based on the prepositions, "in the [time]" and "at [time]" are two different phrases for describing different kinds of time. The word "night" just happens to fit into both groups. Phrases "in" times are more vague, and phrases "at" times are more precise. For example, "at noon" compared to "in the afternoon", where "at noon" is twelve p.m. but "in the afternoon" could be any time from twelve p.m. to maybe five p.m.
Another possible reason gets into linguistic philosophy, but it is possible that "in the night" is not as common as "at night" because of where "in the night" is most often used. The phrase "in the night" appears frequently in artistic literature; in books, stories, and poems. We are used to hearing "in the night" almost only within artistic literature, so it can make anywhere else it is used sound like a story or poem. This can sound strange if the context isn't artistic, but is something else, like making an appointment.