The data is stored magentically on those cards, the cards themselves do not produce any magnetic forces so they cannot affect each other. A bit like storing two old style cassette tapes next to each other. The magnet is on the reader/writer to read/write the data, the actual strips/tapes themselves do not produce any magnetic fields.
Similarly with the chips too, they have no internal power source, there are wires hidden in the cards and when you bring then near a reader, the reader puts out a magnetic field which the wires pick up and induce a current in the chip, powering it and allowing it to be read. As soon as you move the card away from the reader the chip has no power again. It's the same system as door access cards at your place of work.
You can try this with old cards, the other side of the card card to where the chip is, take some scissors and cut half way through it. Your cards chip will no longer be readable as you have cut the wires which induce the current that powers the chip. You may not see the wires if you cut the card in half and try looking for them as they are very thin.
Update: On the card chips side that is used for "chip and pin" payments. Someone with the right equipment in close proximity to your cards can induce the current and then read the data the chip provides. You can get apps on your phone that can ready card data if you have NFC enabled on your phone. I have seen setups costing about £100 and a mobile phone pick up the chip details just by walking past them with dedicated NFC antenners and a suitable power source connected to their phone. If you have "chip and pin" cards there are various solutions to preventing this.
1) Get a wallet that does not allow EM radiation to pass through easily, they cost about £10
2) You can get "interference cards" which you place in your wallet like any other card. They pick up the signal and spew out gibberish data, in effect scrambling the data the cards chip sends out. Again around £10, this is what I use.
3) If you only have 2 chip cards store them next to each other, they will both respond and intefere with each others signals.