❝ I have a right to choose, too. I don't like two-legged things. ❞
For understandable reasons, Nanaki is a little cold towards the party when they first meet him. Consider – he's been locked up for about a year (judging by Before Crisis), being experimented on by Professor Hojo, Shinra's resident mad scientist. With the recent arrival of Aerith into Hojo's lab, he'd locked the two of them together. Hojo, in all his humanitarian goodness, was trying to breed them as both Nanaki is a rare species and Aerith is the last of the Cetra.
I, too, sometimes forget that Final Fantasy VII had suggested beastiality.
Details of what happened while Nanaki was held by the Shinra aren't specified, but there's no doubt it wasn't pleasant. It does seem like he lost an eye while in captivity, as he has both eyes in Before Crisis. And during Final Fantasy VII, Nanaki does mention that he has a score to settle with Hojo. At some point the number thirteen in roman numerals was tattooed on Nanaki as some kind of designation from Hojo. Whatever he went through, we know that it causes Nanaki to question himself later in the game when he wonders to Cloud if he'll go insane like the other tattooed experiments.
So, really, you can't blame Nanaki for being cool towards Cloud and the others when he first meets them. He doesn't even tell them his name, allowing them to call him whatever they wish and generally being polite but distant. At this point, all he's interested in is getting home to Cosmo Canyon. It isn't really until the party get there that they begin to learn more about Nanaki. Up until that point he seems separate from the rest of the party, although still mostly polite and curious about what's going on around him.
It is in Cosmo Canyon that they learn that Nanaki is a lot younger than they expected. Obviously, a human would have no way to tell how old a creature like Nanaki is by looking at them. But aside from that, Nanaki speaks in a polite and formal manner that belies his age. This he does because he wants to be an adult and to prove himself a worthy warrior capable of defending Cosmo Canyon. This is one of the ways he tries to show his maturity. He is impatient for his adulthood to come.
❝ ...grandfather. I want to be an adult. I want to grow up to be able to protect you and the village. ❞
But despite how juvenile Nanaki's attempts to appear grown-up may sound, it does come from a real desire to better himself and protect those that he cares about. Nanaki is courageous, and has a strong sense of honour. And also, despite his impatience in being seen as an adult and warrior, he is a very patient creature himself. He seems to have no issue playing with the children in Costa del Sol when they kick the ball around him. He also has a fondness for Yuffie, despite admitting to himself in Case of Nanaki that she treats him as a pet.
Hello Darkness My Old Friend
But his main reasoning for trying so hard to be seen as an adult? That old familiar friend: Daddy Issues™.
For all of his life, Nanaki believed that his father Seto was a coward who abandoned the tribe when Cosmo Canyon was under threat from the Gi Tribe. This galls him completely, and Nanaki is ashamed of his father. Considering the great pride he takes in Cosmo Canyon and his role as a protector of the place, to think his father ran away like a coward instead of fighting is repulsive to Nanaki. He speaks fondly of his "brave mother", but holds a lot of anger at his father.
Of course, this anger all turns out to be completely unfounded because Seto was actually a hero who sacrificed himself in the tunnels under Cosmo Canyon to prevent a sneak attack from the Gi Tribe. I'm not entirely sure why Nanaki's mother insisted he not be told about what his father had done and why Bugenhagen went along with it for so long (Nanaki is almost fifty years old!). My guess is that because Bugenhagen respected the wishes of Nanaki's mother to not speak about the cave or what happened to Seto, a young Nanaki may have taken this silence the wrong way – they didn't speak of Seto because he was something to be ashamed of. While I suppose we have to get our drama somehow, it all seems a bit unnecessary to me.
❝ Look, Nanaki. Look at your father, at the warrior, Seto. ❞
When Nanaki learns this truth, he resolves to follow in his father's footsteps and become worthy to be the Protector of Cosmo Canyon. He is ready to do so right now, and would sacrifice himself just like his father did if needed, I'm sure. But Bugenhagen knows Nanaki, and he knows that he will hold himself to a high standard – Nanaki will be as hard on himself as he was on his father. He knows that Nanaki isn't quite ready yet.
❝ Ho Ho Hoooo. No Nanaki. You can't stand on your own yet. To do that now would destroy you in the long run. ❞
Bugenhagen also knows that the planet is dying. While he isn't sure if Cloud and the others will be able to do anything, he also knows it's better than nothing. And so, he requests that Nanaki remain with Cloud and see and learn more of the world. This will help give Nanaki the knowledge he will need to be a Protector of the whole planet, and not just of Cosmo Canyon.