To my understanding, in a PoS chain, both forks of a chain retain the staking balances on both sides, and it would only make sense for the stakers to validate the same blocks on both chains.
Sure, but can't the same apply on Proof-of-Work too? Say that the chain splits to old-Bitcoin and new-Bitcoin that supports merged-mining. Miners that mine the old chain can use redundant hashes to mine for the new chain, with the exact same hash rate. All that's needed for new-Bitcoin is to prove you've worked for it. It'll be essentially a sidechain, but with no dependence on the mainchain.
The difference is that there is still "work" being done on both chains, even the merged mined coin is being mined and it could actually have increased difficulty for it to no longer work as a merged mined coin.
In the PoS case, there is no actual "work" being done, and it is trivial for the staker to benefit from both chains at virtually no cost.