Well, it is not always the same process, that's why the problem is so hard to identify.
After windows has been restarted you can usually update Affinity Photo without any problems.
After some time some processes seem to "hijack" the folder (or maybe all folders containing "Photo" or something similar in the name).
On my system it is BrCtrlCntr.exe (Brother ControlCenter 4) in 90% of all cases but not always. On other systems I found other processes in the past.
The problem with the setup program is, that is removes in a first pass all files (which succeeds), then removes the installation folder ("Photo") and in a second pass installs the new files.
The problem is that removing the folder does not work, instead you will get an error message.
The problem can be solved by Affinity quite simply: Either they do not try to remove the installation folder OR they _try_ to remove the folder and when this fails they ask the user if the certain process should be killed (this is done by several programs because that problem is not so uncommon on windows systems).