Wow they have multi-sig support now, that is pretty amazing. Would easily be worthy of its own article in other circumstances.
If I understand it correctly it means money can not be withdrawn without both the account holder and bitstamp signing the transaction. This means that neither can the money be stolen by hackers, nor could bitstamp run off with it. There are actually some bitcoins in the real blockchain "with your name on it", as opposed to you just having a claim on bitstamp.
I doubt that. Having the funds jointly controlled by the exchange and the user, and outside the full control of the exchange, would require each fund transfer to be represented as a transaction on the Bitcoin network, which is simply impossible - the volume of transactions on an exchange is a few magnitudes larger than what Bitcoin can currently handle. To keep up with that volume, exchanges must use an internal off-chain accounting system, which precludes the use of multi-signature (or any other smart contract based on Bitcoin scripting).
In addition, this would require the user to be online and actively sign transactions when a matching order is found.
I think that they most likely meant that they're using multisig internally to protect their funds, in a setup where all the keys are controlled by them.
Not necessarily. Bitgo uses a 2-of-3 scheme, where they provide retail account holders with a primary key, and a backup key. In the usual flow, you and Bitgo sign a transaction. If you need to recover, you can use your backup key.
Source: I am a Bitgo retail customer, but I don't have access to an enterprise account and cannot comment on their Bitstamp integration.
How would they turn on such a feature without each account holder generating keys specifically for those TxOuts and moving the funds into appropriate P2SH addresses?
It sounds more like, it means it's possible given the necessary end-user configuration to enable multi-sig, not that all coins are already benefitting from that level of protection.
Multi-sig transactions work as you describe but many darknet markets have had that option for some time. It's actually recommended you use multi-sig for every darknet purchase.
If I understand it correctly it means money can not be withdrawn without both the account holder and bitstamp signing the transaction. This means that neither can the money be stolen by hackers, nor could bitstamp run off with it. There are actually some bitcoins in the real blockchain "with your name on it", as opposed to you just having a claim on bitstamp.