Byku Is this by design or omission?
I'd say that's by design, but this is just a speculation; 'Incognito' to some extent is a bit 'less customised', some user configurations gets ignored, e.g. non-default 'open with', which in a sense reflects personal usage history, and which other apps is on the device, therefore not incognito-ideal.
Ignoring the custom 'open with' may be a way to make the browser footprint more 'generic'. (I prefer not using the concept of Browser Fingerprint if not precisely the topic, people talk so much about it everywhere)
But again, this is just a guess.