I get that Apple should (and does) ask password when you want to download paid apps. But even in the case of stolen iPhones, why is it important to check identity for free apps ? Is is anyhow related to avoid fake downloads / bots ?
Probably for the same reason you have to enter your password to install stuff on a computer. It's a security layer to make sure only the owner is installing software on the phone.
One reason may be to verify what territory the user is from in order to match it to the developer's permissions. If you have an app that is released only in the U.S. for whatever technical, business, or IP reasons, it would be matched against the downloading user's registered territory (as opposed to current location), which obviously requires a logged-in state.<p>[update] also would help the app store keep track of apps a user already has downloaded.
As well as for security, I believe it's to help with things like purchase history, recommendations etc. technically even though it's free, it's still a purchase. If you change your phone it's easy to back up or remember the apps you had installed by referencing purchase history and stuff.<p>I'm not sure also but if something goes from being free to paid, you are covered to still get it free but that is purely speculative.
Might have something to do with age requirement or age appropriateness - make sure kids can't install apps on mom and dad's phone without permission because they need a password.