DP, Agile isn't a process at all, formal or otherwise. I have a name, and I'm not a process.
Agile is a collection of methodologies and ideologies for managing the software development process - it is a process by definition. Creating software is process. Management is a process.
Hence why everywhere talks about he "Agile Process", e.g. http://www.agile-process.org/
Agile is a modern reincarnation of the 1950s "Iterative and incremental development process" http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1204375 that was redisocvered as the "Unifoied PRocess" in the early 90s.
The fact that there is an Agile Manifesto and that you can buy books on the subject or study it it on courses makes it formal. Doesn't matter that it is adaptive and flexible, it is still a formal process of software development. The process of writing a formal manifesto outlining the principles has formalized the process.
Randomly banging your head off the keyboard is a process (I call it the "process of stochastic cerebral software development").