The word "slot" is confusing in your question and possibly not a big part of what you're after. I think what you are describing has two components. Firstly it is a "latching" mechanism (which is probably a more specific term than "bistable"), in that one press puts it into a certain state and a further press takes it back out of that state. Secondly it is a "catch" or "retention" mechanism, in that the card is retained when it is in one of the states.
In order to replicate within a design it will very much depend on your specifics. If it is purely mechanical then having a sprung tab that drops into a hole when your item is depressed will retain it "latched down", you will then need a method of releasing the tab when it is repressed and preventing it from relocating in the hole as it releases, perhaps by forcing the tab to follow a different route as it returns. Look at the simple mechanisms in various types of ballpoint pen for ways to achieve this. Retaining a card or similar whilst you're in the latched position could just be an extension of this idea, depending on what shape the card is.
If your design is electro-mechanical then you have several further options, perhaps look at the design of toasters for ideas around electromagnets and springs, or CD trays for motor-powered release mechanisms.