How does iOS handle multiple apps registering for the same type of URL? In Android, I get a popup asking which app I want to use for this action, and an option to make that the default for future actions. Is there something similar on iOS?
You get a popup asking which app to use. I don't think there is a "save as default" because that would necessitate a way to manage those defaults, which would expose the underlying mechanism to users, something Apple is generally averse to doing.
The reality is that it doesn't get used a whole lot. Browsers, Dropbox, and other apps of that nature tend to provide the data while miscellaneous apps consume it.