Scrooge gave a dime to young Duke Baloney, hoping that it will inspire the little shoeshine boy to become like him. In a sense, he succeeded, as Duke became uncannily similar to Scrooge, just not the way Scrooge wanted him to be: instead of his hard work and honesty, Duke copied Scrooge's greed, ambition and Scottish-ness. Gone Horribly Right, indeed.
Glomgold being a comparatively recent addition to Scrooge's Rogues Gallery and being a good deal younger than Scrooge has been hinted at since the pilot, where Donald makes it clear that he, a veteran co-adventurer with Scrooge, had never tussled with Glomgold before and didn't know much about him, suggesting that he became a threat to Scrooge in the period between Della's disappearance and Donald's reunion with Scrooge ten years later.note This episode establishes that Glomgold came to Duckburg in the '80s and Season 1 made it clear that the triplets were born between 2000-2009, putting a twenty-thirty year timeline with Glomgold acquired wealth and distinction to rival Scrooge
As pointed out in YMMV, Glomgold asked for a dollar for a spat shine, in a period where that would've been pricier than it is in today's money. Even when he was little, Glomgold was on the path to being a stingy greedy miser. Scrooge just accelerated it.
On the other hand, why was young Duke so offended by Scrooge giving him only one dime, when he hadn't even asked for a dollar or learned Scrooge was "the richest duck in the world" yet? Because he saw Scrooge start to take out a bill from a huge wad of cash tied with a gold money clip, then reconsider and toss him a dime instead. We the audience know Scrooge was reminded of his younger self and wanted to inspire Duke to take the same path he took, but from Duke's perspective, it looked like an obviously rich duck with cash to burn made the conscious decision to short-change him. (Though how he reacts is still Disproportionate Retribution.)
Why was young Scrooge inspired by receiving an American dime, while young Duke was enraged and insulted? One huge difference is who gave it to them. Young Scrooge received an American dime from a poor ditch-digger who obviously didn't have much more than him, making the payment seem fair, while young Duke received a dime from the self-proclaimed "richest duck in the world" who started to give him a fair wage, then visibly changed his mind and tossed him a dime instead, making him look cheap and condescending. (Though how Duke reacted to that rage is still ridiculous.)
Why is Scrooge so enraged at learning that Flinty had stolen the money clip so long ago, to the point of actually accepting a ridiculous bet? Remember his credo: "I'm smarter than the smarties, tougher than the toughies, sharper than the sharpies — and I made my money square!". It may well be the fact that his Create Your Own Villain moment with Duke Baloney galls him, not because of the theft, but because he inadvertently created his own antithesis as well — a billionaire whose entire fortune was built on deceit and theft, the very practices that Scrooge loathes more than anything.
Not only that, but according to Word of God, that money clip contained $2 million. Considering how Scrooge reacts to losing 87¢ in "The 87 Cent Solution", losing $2 million must've been especially infuriating, especially now that he knows Glomgold of all people was the one who stole it.
On a meta level, the reason the previous series' episode where Scrooge and Glomgold fight for a magic lamp was chosen to be referenced when Glomgold mentioned his intention to "Always one-up Scrooge". That competition ended with Glomgold's victory but he ruined it with bad wishing. It shows us that, even when Glomgold wins, he loses.
Young Duke wants to make a drill. Isn't South Africa constantly fought over for oil?
Duke wanted to drill for gold. Is it possible that he was told about oil (black gold)?
Duke's dream: He throws a heart made of flint into a furnace, which gives him coins that begin to overwhelm him. His subconscious is telling him that if he tries to continue life as Glomgold, his money won't make him happy.
Fridge Horror
Young Duke dismisses Scrooge's concerns that stomping coal into diamonds would hurt his (Duke's) feet by saying he's too stubborn to feel pain. This is a description of a very real and dangerous condition called congenital analgesia. If Duke was inflicted with this, it's an outright miracle he survived to adulthood at all, even with Cartoon Physics and Amusing Injuries being the norm for his world. What's worse is that the scene of the amnesiac Duke Baloney casually snapping his dislocated fingers back into place makes it more credible that he does have this trait. Albeit, given he does react to pain at various times, and is described as simply having a extremely high tolerance for pain, it seems he does indeed still feel pain and his remark as a child was simply hyperbole.
Glomgold's backstory in the comics has other reasons to be changed than the timing difference resulting in their ages being different; they still could have met when Glomgold was a young man instead of a child and had things go similarly. However, having a sympathetic character get someone arrested in South Africa, notorious for its horrifying prison gangs, would be... erm.
If Duke was inflicted with this, it's an outright miracle he survived to adulthood at all, even with Cartoon Physics and Amusing Injuries being the norm for his world. What's worse is that the scene of the amnesiac Duke Baloney casually snapping his dislocated fingers back into place makes it more credible that he does have this trait. Albeit, given he does react to pain at various times, and is described as simply having a extremely high tolerance for pain, it seems he does indeed still feel pain and his remark as a child was simply hyperbole.