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.
Yes but the User Experience on a computer is not the same at all. On a real keyboard, typing is much faster and therefore the pain is much smaller. Mobile is on the go, usually apps are designed to make everything easier for the end user, and Apple is usually good at doing exactly that. Except here.
And also apps aren't necessarily sandboxed on your computer, so you could damage your computer by installing new software. (Which is why you want to make sure the owner is installing it.) You can't do that on an iPhone anyway, so it seems like identity check is useless.