Does iOS allow any app to look at the clipboard data?
The whole purpose of clipboards is that apps can read from it and write into it. Usually this happens due to a copy/paste request by the current users, but there are apps (e.g. Podcast apps or browsers) which peek at the clipboard to see whether it is in a any form actionable by the app (e.g. if it is an URL). And there are, as recent news show, apps which seem to peek into the clipboard to grab data, passwords etc.
If you are curious about how code to access the clipboard can look like, https://stackoverflow.com/questions/41452868/reading-from-the-clipboard-with-swift-3-on-macos gives some examples.