The underlying structure of Mac OS X was adopted by iOS, however the implementation provides a fairly radically different user experience than OS X. Over time, features and design idioms are cross pollinating with many features from iOS finding their way to OS X beginning in Lion v10.7, culminating in Yosemite with APIs that allow handing off editing documents in supported applications between Mac and iOS devices.
Once it became clear that iPhone OS and iPod Touch OS were too unwieldy, the term iOS was born. iPhone, iPod touch, iPad and Apple TV (from version 2 forward) all run on the same core OS, though functionality is much more limited on Apple TV. The sharing of the core architecture allows functions such as airplay to work with little to no configuration required by the user.
Each device gets a customized build even when the major build numbers (e.g. iOS 5.0.1) are the same and the software is released and documented as one build.