Long answer: The beacon chain _will_ run in parallel until the two chains are merged. Until that time, ETH in the Beacon Chain isn't transferable, so effectively not a separate asset.
However, exchanges are offering Eth2 staking derivatives which they're branding as "ETH2". But it should be remembered that this is a derivative, not a separate M0 asset.
At any point in time, of course, any node or set of nodes can declare a fork in the chain to be invalid and reorganize on another chain. That's what happened with Ethereum Classic and it could happen again. I think less likely unless there's a vulnerability discovered in Eth2, though.
Also less likely because in some ways the Ethereum Classic incident helped cement Ethereum's place by scaring off all the ideologues and keeping the participants who were more pragmatic.
Check out https://ethereum.org/en/eth2/
"What is Eth2?
Eth2 refers to a set of interconnected upgrades that will make Ethereum more scalable, more secure, and more sustainable. These upgrades are being built by multiple teams from across the Ethereum ecosystem."