Is there a similar feature in Safari Web Inspector?
I don’t know when this was added, but it is present in Safari on El Capitan (Safari 9.0). It works the same as in Chrome (right click in the console window and select “Keep Log on Navigation”).
Update: As per Daniel Compton’s answer, in Safari 11+ this is now under the settings icon as ‘Console: Clear when page navigates’.
Update: The setting is now back to “Preserve log” in the Network tab in the developer console in Safari 14+