Angst? What Angst?: Tweed drops Tod off in the forest minutes before a rainstorm, leading to the most miserable night of his life. The next morning, he gets one look at Vixey and forgets all about those pesky abandonment issues. Even when we see him forlornly looking down from a hill at the end of the movie, it's framed more as him missing Copper than the woman who raised him.
Applicability: Two innocent young kids become friends because they're oblivious to how the rest of the world sees them, and then grow up into the roles society expects of them and are torn apart. There's also these lines from the song "Best of Friends":
Broken Base: Disney fans are split between whether this is a heart-wrenching childhood classic or the red flag of Disney's Dork Age finally hitting an all-time low.
Family-Unfriendly Aesop: It basically says The Power of Friendship and The Power of Love can't overcome hard-wired reality. Also that racial differences cannot be overcome and that everyone's just better off ending their friendships and living apart from one another with their own kind. Nice going, Disney; this is one of the rare instances where they didn't attempt to use either classic trope to get a fully happy ending.
Just Here for Godzilla: There are actually some people who only watch the film for the Dinky and Boomer scenes.
Moral Event Horizon: Amos Slade comes dangerously close to crossing it when he attempts to shoot Tod after he saved their lives, but manages to avoid it when Copper convinces him that he's not their enemy.
Shipping: Given the fact that their relationship drives the plot, it's not surprising that Tod and Copper get their share of this. Even better, it takes the above mentioned Applicability to a whole new level...
Tastes Like Diabetes: A common misconception of the film due to how it's advertised. While there are a few overly cute scenes during the eponymous duo's childhood, the movie is actually taken very seriously and is much more adult-appealing than most people seem to realise. Played more straight for the Denser and Wackier Interquel.
Many fans think Chief should have been Killed Off for Real to make Copper and Amos Slade's anger more justified and overall have made the latter half of the movie a lot darker. It turns out that the writers were going to kill Chief off (as in The Fox and the Hound the novel but decided against it because they thought it'd be too dark. Chief's original fate is apparent when Copper finds him after he's jumped off the bridge: a dog lying motionless with his nose in the water is pretty much dead. But Disney decided to draw the line there.
When Copper returns from his hunting trip with Amos and Chief, some fox pelts can clearly be seen on the pile of fur he's helped collect. You'd think it'd be an interesting/important moment to show the first time he had to chase a fox and/or watch Amos shoot it, but we never see it.
Tod's mother is killed when he's only a baby, he gets chased and shot at by Chief and Slade, respectively (twice!), Copper disowns him and wants him dead, Tweed drops him off in a wildlife preserve for his own safety leaving him without survival skills and a friend in the world, not to mention his neighbor is a cantankerous Bad Ol' Badger. Things get a lot better after he meets Vixey, though...then worse...then better.
Tweed as well - it's made very clear that Tod is her only source of companionship, and that it's just as heartbreaking for her as it is for him, if not more so, when she has to Shoo the Dog to protect him.