I doubt there's an answer to this beyond "because that's how Google designed its apps". Android itself does not require a Google account to use, only Google's proprietary applications do. CyanogenMod and other pure source builds of Android are actually forbidden from distributing the Google apps within the firmware package itself because of their proprietary nature, and if you install such a ROM without the associated Google apps add-on you won't even have the ability to add a Google account in your settings.
If you don't want to be tied to Google then don't use the apps they produce, but also don't be surprised that they want you to sign in when using services that they provide. It's really not that different from any other app or website that requires sign in to use the services it provides, in my opinion, but if you don't like it use different apps.
Re: the question about why Maps specifically requires a sign-in on Android but not iOS: the comments about the iOS one being developed by Apple could certainly be correct, and I do agree with those statements. In addition, Apple may not want you to sign in Google's servers with their Maps app. It would create a tie (however loose it may be) between you and Google's services, which I'm sure Apple is trying to discourage as much as possible to prevent customers from jumping ship to Android.
Another angle you have to keep in mind is that Google is pushing pretty hard to tie together all of its online services as much as it possibly can. Based on their priorities and their business model, it makes sense that the Android version would try to integrate as tightly as possible with their other services and with the web. It's in their best interest (from a business perspective) to try to make Android apps and their web counterparts as seamless as possible in order to attract users to Android who are already heavily invested in Google's products. Of course, this is just conjecture on my part, I think only Google could truly say why Maps works the way it does.