Every device you wish to use with Apple Pay has to have the card loaded onto it, there’s no setup syncing between devices. In System Preferences on the Mac there’s a ‘Wallet & Apple Pay’ setting where you can add the card. When I was setting it up the only option it offered to activate my cards was to call the BNZ. I haven’t had a chance to see if it’s working yet.
I don't have the Wallet & Apple Pay option in my system preferences but my understanding of Apple's support page is that this is only available on Macs with integrated Touch ID. It appears that Macs without this feature can initiate an online Apple Pay transaction, with the transaction completed on an iPhone.
You're right, I wasn't aware of that.