In short, the chances of two people generating the same private keys are so low that it is not worth worrying about. Do you worry about being hit by a meteor?
Ok, but that doesn't really satisfy. Because as more people used bitcoin, and more an more addresses are generated, and the longer its been around, there would be more and more of a chance of duplicate addresses.
Question: after an addresses is loaded with funds and added to the blockchain, then, does that information get sent to these automatic addresses generators in all the web, desk, and mobile wallets, and the bitaddress.org generator too? I don't think so. So how do they know they are not creating an address which was already created?
The numbers are so huge that none of the things you mention, even multiplied by a billion, make the slightest difference.
The number of possible addresses is 2
160. If 10 billion people have generated 1 billion addresses each, then the chances of a new address colliding with any of those is 1 in 2
141. That's a 1 in 2,787,593,149,816,327,892,691,964,784,081,045,188,247,552 chance.
Let's pretend that Bitcoin is as big as Visa, which can process 24,000 transactions per second. In a million years, bitcoin will have processed up to 800,000,000,000,000,000 transactions. If a new key were generated for each of those, then there is still only a 1 in 2,000,000,000,000,000,000,000,000,000,000 chance that it will be a duplicate.
people having generated the same key is bigger, and that is needed for collision. The chance is still low enough to be neglected. But to answer OP, yes it is possible that the next time you generate an address with your client, you get the key to Satoshi's main wallet and can use his coins. But the chance is really small. Not like winning the lottery, more like the event that you win the lottery this week and your brother wins it the next week and then the both of you get shot while shopping for groceries in two seemingly unrelated killing sprees and then the killers share the next week's lottery jackpot and Macarena regains its popularity and becomes the numbers one hit on major radio stations, and before that a bitcoin whale reading this message thinks it was the best and most inspiring thing he ever read and contacts me and sends me 500 bitcoins because he was so impressed, you know, that sort of thing.