Online it says that the BakıKart has no expiration date and allows you to pay for multiple people.
So does that mean we can buy just one card for two people? And how does it actually work in practice when paying for two?
Yes, one card is enough. You just tap it for each entry until the balance runs out. 
So you only tap it at the entrance? You don’t need to tap when exiting?
No, only once at the entrance — there’s a terminal there.
For example, in the metro: one person holds the card in their right hand and taps it once on the terminal. When the light turns green, the second person goes through the turnstile. Then the person holding the card taps it again, and they pass through the turnstile afterwards.
At the exit, both of you just leave the station freely since there are no barriers.
On the bus, you simply tap the card twice on the terminal near the driver as you enter — and that’s it.
Can you advise where do I get this transport card?
The transport card is sold at the airport in special machines — the same machines where you top it up are the ones where you buy it. In the city itself, it’s sold in metro underpasses, also in vending machines or at the ticket office. Here is what the machines and the card look like.
If you buy the card from a machine, be aware that the machine does not give change. If you insert 5 manats, then 3 manats will be added to the card balance, and 2 manats will be deducted for the card itself.
You can also use the card to get from the airport to the city center. The shuttle bus runs every 30 minutes from 6 AM to 1 AM, departing from the airport to the city center. The fare, if I’m not mistaken, is 1.30 AZN per person. One card can be used by several people without any problems — the only thing that matters is that the card has enough balance.