I purchased Affinity Photo last year (thru the Apple Store) and recently updated to 1.7.1. Downloaded Publisher from the Affinity Store yesterday (June 19). Then purchased Designer from the Apple Store as well.
Launched Publisher and noticed that Studio Link would work for Designer, but not Photo (message says I need to install the program).
I found a simple fix. Launch all three programs separately from the icon (I'm on a Mac Pro), then check to see that Studio link is operating properly.
Close everything and relaunch Publisher only. Studio link should now communicate with the other personas.
No re-installing of programs or esoteric terminal mode shenanigans required.
Hope this works for you.