The app itself isn't expensive - it's the time and resources to connect with local governments (so police search the database too) and the expense to reach more shops and integrate with various point-of-sale systems in bike shops.

The idea of the police engraving anything into my bike is horrifying. And impossible for carbon bikes.

And once again, it won't be difficult to get registration for free. The reason that there is a fee is to encourage people to register their bike when they are in a bike shop rather than putting it off because they can just do it at home - plus, by adding a fee we will make it easier to track thieves who register bikes they've stolen.