For as long as they have shared the panel together, Batman and Nightwing have usually been presented as having a father-son relationship.
When he first made his comic debut as Robin, Dick Grayson was Batman's Boy Wonder, his trusty sidekick ready to leap into action at a moment's notice. Those father-son parallels seem all the more apparent when considering that their relationship began when Batman legally adopted the boy following the murder of Dick's parents, quite literally becoming his surrogate father. Even when Dick went on to become his own man as Nightwing, he still found himself leaning on the older, wiser Batman for guidance in extreme times of hardships, much like a son would a father at any age. It's all the more apropos to see Dick essentially follow in his adopted father's footsteps when the original Batman is presumed dead after Final Crisis. After winning a Battle for the Cowl, Dick Grayson officially becomes the new Batman.
Feeling obligated to his seemingly deceased mentor, Dick jumps at the opportunity to try to revive Bruce Wayne in a Lazarus Pit, going into Batman & Robin #7 by Grant Morrison and Cameron Stewart. When questioned by Batwoman, Dick explains himself, as well as why he's the new Batman, by talking about his relationship with Bruce. Interestingly enough, he doesn't refer to him as a father figure, but instead as a "brother" and "best friend." The idea of Dick and Bruce being surrogate brothers rather than father and son puts their relationship in an entirely new context. For better or worse, Dick viewing Batman as more of a brother drastically changes their dynamic, as well as re-contextualize previous interpretations of their bond.
Batman is Nightwing's Surrogate Brother, Not Father
For instance, Batman fires Dick as his Robin when he's nearly killed by The Joker in the line of duty way back in Batman #408 by Max Allan Collins, Chris Warner, and Mike DeCarlo. Bruce lets him go, not feeling comfortable putting his ward in harm's way anymore. This initially feels like a concerned parent trying to protect their child in their own complicated way. Dick even seems to receive this as such, furious that Batman views him as a child. However, in now viewing this through a more brotherly lens, readers can see how Batman firing Dick as Robin hits different when Dick is being rejected not by someone he looks up to, but instead by an equal. Nightwing isn't looked at as a child just by the man who raised him, but in his eyes being belittled by someone who he reveres as a colleague, which might be even more gut-punching for him.
By becoming Batman, Dick isn't metaphorically or literally stepping into his father's shoes as much as he's promoted to a captain's role when his co-captain has fallen. Batman and Nightwing being figurative brothers, partners in crime-fighting so to speak, forces audiences to view their past and futures in a new light that in many ways brings the two even closer than they were before.