Can I perform evil twin attack with my WiFi card and Kali Linux PC or do I need an adapter for the attack?
A wireless network card can act as an access point if the card supports "master mode".
If you run
iw list, check to make sure "AP" is in the list for that interface. This should be the case for most cards.
This also has nothing to do with monitor mode, since monitor mode is a passive mode and advertising a network requires transmission.
With only one card, you can spawn an evil twin access point. So the victim will try to connect to you and see that there is no Internet connection available with your AP..
But some advanced attacks consist of use the first card to emit the fake AP and connect the second card to the real AP. In this configuration, you can forward the victim traffic to the real existing targets and spoof what you want (through DNS fake response for example).