I've entered my passport number to airline apps (e.g. Ryanair), accommodation apps (e.g. Airbnb), ride-sharing apps (e.g. Blablacar), visa application websites (e.g. the agencies that processes visa applications for Russia, China, Nigeria, Vietnam etc.), and car rental websites. I've written it in big books that could be scanned by anyone who arrives at an albergue with a smart phone. And I've shown my passport to countless airline check-in agents, hotel receptionists, post office clerks and so on - some of whom have taken the passport away and made a copy of it.
From that list, the ones that worry me the most are the visa application agencies. They collected a ton of information and I paid with a credit card ... and in the case of the agency that processes Russian visas, I worry that they might have connections to organized crime, and I know they have connections with the Russian state, which is basically the same thing.
All of which is a long pre-amble for a few thoughts:
1) It does seem kind of excessive to require a passport number to use an app
2) If criminals could commit identity fraud with just a name and passport number, I think it would be big news.
3) I trust the security of the Alert Cops app against hackers at least as much as I trust my airline's app
4) Maybe you can enter a false number and the app will work?