Of course they need a passport to check your identity to collect the money, but why they ask for the visa status too? It's their concern? Means that without a valid visa, you are not able to pick up the cash?

Without a valid visa, you are here illegally so it's possible they are being asked to check since people here illegally are more likely to be involved in money laundering and other criminal activities. Or maybe the person you dealt with was just being nosy and it's not a universal policy.

It's relatively new, but yes, apparently financial institutions have been instructed to check a foreign customer's permission to stay when conducting business. There have been reports of this happening before. 

Yes, totally correct! They are instructed to look at permission to stay. If you are here illegally, you neither have the right to use services in Thailand.