Could it be that the mobile interface was simply better in terms of speed, usability, smoothness of the UX? Complex web apps honestly often look and feel shite in the browser.

Nah, people retain on mobile apps because they have a simple icon to use to get there. It’s extra effort to remember the name of the service and write down the address.

Push notifications are really powerful for creating habits as well.

Potentially, but aren't those essentially properties of native apps? When we say that apps retain users better it's not some magic, it's because of apps typically having better UX, whether that's speed/features, device integration, or the "UX" of being right there on the home screen.

How do you know that people use the app because they are already highly committed to the service, rather than the other way around? IE, the whole thing is a selection effect?

We were a logged-in only service, and measured that users acquired on the app who never used the website were typically retained longer than users acquired on the web who never used the app.

We certainly did encourage users acquired on the web to download the app, and the fact that those who did were retained longer could have been a selection effect, but I don't think this accounts for users who never saw the other platform.