Custom Colors can be synchronized among different iOS devices by enabling Synchronize Colors.
In the app go to Settings > General > Synchronize Settings.
To be able to do this technically, app needs to be able to store the color somewhere in the event that will get synced to your other device(s).
The only place available for this is the notes field of an appointment.
When you enable ( Settings > General > ) Color Syncing, the assigned custom color will be stored as a small piece of text in the notes of an event.
This piece of text will be automatically hidden inside the app, but will show up in other applications that don't support color syncing.
Due to technical limitations syncing colors with Outlook is not possible.