Can a food contain 'extra' sodium?
I think you've misunderstood people talking about sodium; I would expect they are generally just talking about table salt, NaCl.
Addressing your question about quantity, it's very common that when cooking, people judge ingredients by eye and by their expectation of what a normal quantity would be, so use much less salt than a restaurant or commercial preparation. (Something similar is true of butter.) Commercial ice cream, for example, will contain salt, but home ice cream makers are unlikely to include much if any.
If you want to think of that as 'hiding' sodium in products, I suppose that's reasonable, but it's not being done through mystery sodium compounds you're not aware of.