Other POS tokens do not require the wallet to be open all the time to stake. You get the same return whether the wallet is open or closed. Of course you eventually need to open the wallet for a while to receive the stake. In other words the stake you receive is proportional to your coin age rather how long the wallet has been open.
Is it the same for BCS/LCS?
Bitcoinstake is not an ERC-20 token running on eth blockchain. That's the reason why those tokens need not be online since it's just relying on eth block creation.
Coins which have their own blockchain, requires wallet to be online and open 24x7 since they are acting as nodes supporting the specific coin's chain and get rewarded by staking their coins.