As this problem in principle is the same problem as last year, when we wanted to run Xcode 12 on macOS Monterey, it is worth to check last year's question on the same problem. There, I found this great answer in which a script is proposed that only needs to be run once to fix the problem (allowing a regular opening of Xcode 13, e. g. via double click). The script works by changing the build version of the old Xcode 13 to the build version of the new Xcode 14, thereby tricking the OS.