Not
always true. The fighter community has a lot of combat formations that have aircraft flying quite a long ways away from one another. In the USAF, Tactical Formation can have two aircraft flying between one and two miles away from each other, but still maintaining a precise position (in the case of this photo, line-abreast position, but there are others, too) and with the wingman station-keeping and maneuvering in precise relation to the flight lead. Tactical is the primary "fighter" formation, as it maximizes forward sensor/weapon firepower, while allowing better visual lookout between the two, denying "tally two" to adversaries, and allowing individual maneuvering once tactical execution is in progress.
Flying tactical formation is actually quite a bit more difficult than flying "close" formation (ergo, fingertip, etc) because you have to rely on more subtle perceptions of visual cues of being out of position. There is obviously less danger of a midair if you screw up, but the chances of an angry instructor killing a poorly-flying student are significant. We don't jack around when it comes to flying good tac form.
Here are two T-38Cs in line abreast tactical formation. The briefed position is 6,000' line abreast, but for this photo I had the wingman collapse in to 4,000' because it made for a better photo.