Perhaps in this fanfiction, Naruto is in a situation where he has no other source of sustenance. He could be trapped in a desolate place with no food or water, and the blood becomes a last - resort option for survival. This would be a very dark and extreme take on the character, as it goes against his usual values.
One possible reason could be for power enhancement. In some fanfic concepts, drinking blood might be associated with gaining special abilities or strength. For example, if there's a dark magic or forbidden jutsu involved that requires the consumption of blood to unlock certain powers.
Physically, he might gain new powers or abilities as mentioned before. But there could also be negative effects. His body could reject the foreign blood, leading to internal pain and weakness.
One reason could be extreme emotional pain. Maybe he witnesses the death of a very close friend or loved one in the fanfiction world. This overwhelming grief could cause such a physical manifestation as crying blood.
Maybe he was poisoned. There are many villains in the Naruto world who use poison. If Naruto was poisoned during a fight or an ambush, it could lead to him coughing blood as his body tries to fight off the toxins. Just like how Sasuke was once poisoned and had to be treated.
One possible reason could be some sort of dark curse or forbidden jutsu. Maybe Naruto was the victim of a malicious experiment that altered his blood composition. Another reason might be related to his connection with a dark entity. For example, if he has some link to an ancient evil, it could manifest in his blood turning black.
One reason could be overusing his chakra. In Naruto's world, chakra is like life energy. If he overexerts himself and drains too much chakra, it can cause internal damage, leading to coughing up blood. For example, when he was training intensively or during a tough battle where he pushed his limits.
Well, it's a very strange concept. One possibility could be that Naruto gains some sort of quantum - like abilities. Maybe he could exist in multiple states at once, like being both here and there in the ninja world, which would give him an edge in battles as his opponents wouldn't be able to predict his exact location.
Maybe he discovers some dark secrets in the Blood Prison. These secrets could be about the corruption of the ninja world or some hidden evils related to powerful people. This new knowledge might make him lose faith in the system he once believed in, leading to his transformation into a darker version of himself.
He might be trying to deal with stress. Naruto has a lot on his plate, like protecting the village and his friends. Sake could be a way for him to unwind in the fanfiction.
One possible consequence could be that Naruto is ostracized by some in the ninja world. Since controlling blood is a rather strange and perhaps 'creepy' ability to some, they might view him with suspicion or fear. This could lead to him having to prove himself more often in battles and interactions.