That’s how it should be. WhatsApp’s desktop apps are technically another front-end to your secured database on your phone. I’m searching for the tweet of one of iOS security researcher that confirmed this. That’s also the reason that your WhatsApp desktop client stops working when you are out of range. That’s being said, I’ve noticed that recently this is changing and somehow the desktop app maintains the data a bit longer. It’s still not completely independent but I think it’s in the transition period. If that happened, then their E2E has to be different.
By the way, I think the security code from one of your devices, probably your phone, can be encoded to the QR code and transfer to the client.
I assume WhatsApp implementation is a hybrid between mirroring the information and some sort of secure handshake via the QR code.