There are quite a few such fanfictions out there. You might find one on popular fanfiction websites like Archive of Our Own or FanFiction.net.
Honestly, it could be any number of fanfictions. You'll have to do some digging on fanfiction hubs. Maybe start by looking for tags related to both Naruto and Sesshomaru and see what pops up. Chances are, you'll find something that fits the bill.
Sesshomaru would likely be aloof and haughty at first. He might look down on Naruto as a mere human. Naruto, being his friendly self, would probably be curious about Sesshomaru and try to start a conversation. Sesshomaru might respond with cold remarks, but Naruto's persistence could gradually break through his icy exterior.
One common theme is power. Since both Naruto and Sesshomaru are powerful in their own ways, fanfictions often explore how Naruto would handle Sesshomaru - like powers or how he would adapt to new abilities. Another theme is identity. Naruto has his own strong identity as a ninja, and when combined with Sesshomaru - like traits, it creates an interesting exploration of who he becomes.
Well, one unique aspect is the combination of two very different mythologies. Naruto is based on ninja and Japanese folklore related to that, while Sesshomaru is from Inuyasha's world which has its own set of yokai and feudal Japan elements. When combined in fanfiction, it creates a whole new and exciting world for the readers.
There are definitely some out there. I've seen ones where Naruto is put in a situation similar to Sesshomaru's, like having to deal with a powerful enemy or a complex moral dilemma in a way that Sesshomaru might. It's really interesting to see how authors blend the two very different universes of Naruto and Inuyasha (where Sesshomaru is from).
There probably is. Fanfiction writers often come up with very creative and unexpected crossovers. However, the quality can vary greatly. Some might be really well - written with interesting plotlines that blend the Naruto and Inuyasha universes in a unique way considering the relationship you mentioned, while others might not be as good.
One possibility could be that in this fanfiction, female Sesshomaru uses her demon powers to protect Naruto. Maybe she has her own motives, like she sees a kind of potential in Naruto similar to what she once saw in someone else. She could be a very strict but effective guardian, teaching Naruto new combat techniques from her own demon - related knowledge.
In this fanfiction, Naruto raised by Sesshomaru could lead to a very different Naruto in terms of personality. Sesshomaru is cold and aloof, and while Naruto has his own warmth, he might become more reserved under Sesshomaru's influence. His view on friendship and loyalty could also be shaped in new ways as he is exposed to Sesshomaru's way of life.
In a 'naruto adopted by sesshomaru fanfiction', the plot could start with Naruto being exiled from Konoha for some false accusations. He then stumbles into Sesshomaru's territory. Sesshomaru, initially indifferent, realizes Naruto's strong will which reminds him of his own. So he adopts Naruto. Naruto then has to adapt to the new world of demons and yokai, while also trying to find a way to clear his name back in his own world.
One interesting 'naruto sesshomaru fanfiction' could be a story where Naruto and Sesshomaru team up to fight a common enemy from both their worlds. Maybe an ancient demon that threatens both the Naruto universe and Sesshomaru's feudal Japan - like setting.
It could be really wild. Naruto's energetic and friendly nature would contrast with Sesshomaru's cold and aloof personality. Maybe there would be a story about how Naruto discovers his connection to Sesshomaru and has to navigate the Inuyasha world with his ninja skills.