Google is similar in that you can have multiple devices synced to one Google account. The difference is that your Google account is not just your music/video files and apps--it's your email, calendar, docs, etc. As far as I know, iPhones and iPads don't have something like an "iMail" app that is limited to syncing with some kind of Apple-only email account, akin to Gmail--they have an email app that you can use to sync with whatever email services you have, which is just like the Android native Email app.
And with Android, you're not burdened by a bloated clunky program like iTunes (at least, it's bloated and clunky on PCs).
The solution that MJKearney and I suggested of making a new family Google account and syncing all devices to that account is very similar to having one iTunes account with multiple devices associated with it. You just have to accept that you can't use the Gmail, Calendar, or People apps to directly access your own personal information--however, you can always use the native Email app to connect to your Gmail account, and I'm sure there are tons of 3rd party calendar and contacts apps in Google Play that can access your personal account info on a device that is registered to a family account.