Can someone living in N.Y. use their food stamp card in Cal. or some other state?

It depends on the company that has issued the EBT card. Even though Food Stamps is a federal program, the government contracts with different companies to issue the cards in certain areas. To give an example, Missouri and Kansas uses different EBT card issuers. The Kansas network accepts EBT cards from the Missouri network, but Missouri does not accept cards from the Kansas network. So you can use your Missouri food stamp card in Kansas, but you cannot use your Kansas card in Missouri, or at least it was that way a few years ago when I worked for the Food Stamp office.

It depends on the store. The grocery store I used to work at would take some states at the cash register and others had to be run at the customer service desk on a different machine.

Yes. Food stamps is a federal program.