If you need emergency cash, do they send you some sort of code and you get the money from the ATM without any card? Or, do you have to show your ID at a local partner of the service to get cash? You would be screwed if you lose your ATM cards and IDs, which could often be the case.
The general process is that:
You contact your scheme (eg Visa, Mastercard, American Express) and advise that you need emergency cash (because your card has been lost, stolen, damaged, etc). At this stage you provide certain information to them, commonly name, date of birth, address, SSN (if applicable), mother's maiden name and so on.
Your scheme will transmit this information to your issuing institution (if applicable), who will then have their own processes for verifying that you are who you say you are. They then either approve or decline this request.
If it is approved, your scheme will contact a wire transfer company - usually Western Union - and arrange for the emergency cash to be provided. They will separately give you information like a control number, and provide Western Union with information that you can verify yourself with in the event that you have lost your ID. This could be secret questions, for example.
Alternatively, you can also, subject to your issuing institution's approval, authorise Western Union to release the funds to someone else. This could be someone you're travelling with who still has their ID, for example. Again, this depends on whether your issuing institution will approve this, but I've approved such requests in the past.
So, to address your question in brief: very rarely do you obtain the emergency cash from an ATM, generally you will obtain it from a wire transfer company; in the event you lose your ID, you still have options.