The premise of the question is not correct, but there is a general shape to rivers.
From Leopold and Langbein, writing in Scientific American:
A sample of 50 typical meanders on many different rivers and streams has yielded an average value for this ratio of ahout 4.7 to one.
The ratio they use in that article is different from the definition you (and Wikipedia, and comments) are using. I'm sure it's simple algebra to convert one ratio to the other. But that article also notes that:
For the large majority of meandering rivcrs the value of this ratio ranges between 1.3 to one and four to one.
Whatever the conversion comes out to, there appears to be quite a range of real-life meander ratios.
That Scientific American article is a summary of a longer, more technical article by the same authors. The method they use is to fix beginning and end points A and B, and allow the river to random walk from A to B. The most probable shape for such a path is what they call a "sine-generated" curve. At a given point, the angle between the tangent to the river and the mean direction of the river the sine of the distance along the channel. The resulting curve is not quite a semi-circular curve, so the meander ratio is not predicted to be $\pi$.
A more recent study by Garret Williams confirms Leopold and Langbein's results, and reports that the most common value for the ratio of the radius of curvature to the channel width is between 2 and 3.
The other major effect driving river shape is how easily the river can erode the soil that it passes through. The river will tend to flow more directly downhill if the surrounding soil is difficult to erode. In areas where the soils erodes quite easily, the river will assume this sine-generated curvature. As an example of that, you can look at the Mississippi River in the United States. It has the classic sine-generated shape all the way from about Cairo, IL south to New Orleans, LA. But it's much straighter up along the Illinois/Iowa border.
