I have both Photo and Designer, purchased via AppStore, the latest available versions. Publisher doesn't seem to be able to link to them. I tried restarting all 3 apps several times. Please check the screenshot.
To enable those buttons you MUST be using version 1.7.1 (or above) of Affinity Designer and 1.7.1 (or above) of Affinity Photo (and each of them run at least once). These builds are now available from whichever store you bought the software from.
For the Mac App Store it sometimes takes a few days to roll out new builds to all users. but here is our advice to see if it is ready for you.
For the Affinity Store