first of all What is extensible design?Is it a software architecture or a design approach in software?
Extensible design in software engineering is to accept that not everything can be designed in advance. A light software framework which allows for changes is provided instead. Small commands are made to prevent losing the element of extensibility, following the principle of separating work elements into comprehensible units, in order to avoid traditional software development issues including low cohesion and high coupling and allow for continued development. Embracing change is essential to the extensible design, in which additions will be continual. Each chunk of the system will be workable with any changes, and the idea of change through addition is the center of the whole system design. Extensible design supports frequent re-prioritization and allows functionality to be implemented in small steps upon request, which are the principles advocated by the Agile methodologies and iterative development. Extensibility imposes fewer and cleaner dependencies during development, as well as reduced coupling and more cohesive abstractions, plus well defined interfaces
And Is implementation with this method common?
It is a design approach. Basically, decompose your work so that it is understandable by human beings as well as the computer. For example, use OOP, useful variable names, comments, and well-named methods instead of writing monolithic functions with variables all called x and y. It is the core of modern software engineering, and allows other humans to read and understand your software (including yourself six months later).
Everyone has written non-extensible software, scripts that you write once for a single execution and plan to never use again (and generally hope that nobody ever sees, because it is slightly embarrassing). That isn't software engineering, it may be functionally "correct", but the lack of extensibility means that it is effectively useless the next time it needs to be changed, and likely will have to be rewritten.