The world has Fibonacci-like properties all over, not because we choose to view it through that lens, but because it's such a simple and ubiquitous quality of reality. In many cases it's easy to see how "the new value is the sum of the two previous values" is a concept that describes basic processes. You would be hard pressed to find a more direct description of those effects.
Exactly.
The reason Newtonian physics is so successful is because almost anything can be described by a second order differential equation. The world behaves as simple as that and it would be quite weird if it required a larger order to describe it.