From the point of view of Bitcoin, Bitcoin didn’t split; someone made a copy, so the initial balances are inherited from the Bitcoin blockchain.
No, that is not what's happening here: Bitcoin is a consensus network where consensus is achieved essentially by mining sequence of blocks, which requires a huge amount of computational power. Each block have a relation with the previous block. Sometimes two or more blocks are mined almost at the same time (creating a fork), miners then choose one and keep mining as fast as they can. At some point one fork will be longer than the other/s, which is when consensus happen: miners abandon the shortest forks and keep mining on the largest one.
I had a company I have shares in that split in two and they gave me a choice, either convert my shares to the new company or keep them in the existing company. I didn't automatically get shares in the new company and if I opted too I would have lost the shares in the original company.
Was this a publicly traded company?
Edit: usually (always?) the exchange is preceded by a partial IPO (or a stake in a quoted company exists for some reason) and they are distributing the remaining shares.