Short answer: no, there's only one Ethereum and one ETH asset
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.
Well, the expectation is that compliant clients will only respect transactions on Eth2 at some point and will no longer consider new blocks on Eth1 valid.
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.
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."
So instead of the promise of a stable, ever-deflating currency value that was made by bitcoin proponents we get unbounded inflation instead? Nice!
Edit: might not be the case for Eth2, but it still seems to ring true for the cryptocoin ecosystem in general: as soon as the value of one currency rises enough, alternatives will be created.
A very, very concise brief is it's a concurrent (happening at the same time) network upgrade from PoW (proof of work) to PoS (proof of stake). The new chain called the beacon chain will operate alongside the "old" one until a full cut-over happens some time in the future. Predicted 1-2 years.
