I'm very interested to learn about historical phenomenology of proportion and ratios, i.e. I want to know how and why people in the past introduced these concepts. As far as I understand symmetric objects or patterns in geometric figures found in nature suggest the idea of proportion and ratios to human mind. But can someone explain how exactly they came to be studied with examples and details?
It seems that originally ratio and proportion emerged not so much "from nature", as from human activities, first practical and later more theoretical. We often find things in nature only after we already know what to look for. I will give several early examples.
In construction of the pyramids ancient Egyptian builders needed to maintain constant slope. Special measure called seked was introduced for this purpose, which measured it as rise over run. The use of seked implies practical understanding of the fact that sides of similar triangles are in the same ratio.
According to tradition the same fact was used more consciously by Thales of Miletus to measure the height of a pyramid, and determine the distance to ships at sea, described here. Although the attribution may be anecdotal it shows that Greeks were aware of such methods already in 5th century BC.
According to Xenocrates (4th century BC) Pythagoras noticed that different sounds appear in consonance to us when the lengths of strings producing them are $2:1$, and in dissonance when they are $9:8$, etc. Attribution to Pythagoras in this case is very likely fabricated, but Pythagoreans did develop a system of musical tuning based on such observations. This one is the closest to coming from nature, but still involves instruments.
The famous golden ratio, that some find everywhere nowadays, does not seem to have emerged from direct observations of nature, but rather from Pythagorean obsession with "mystical figures". Namely, lines in a regular pentagram cut each other in the golden ratio. The idea that the ratio represents "ideal proportions" of the human body was introduced into sculpture by Phidias only after the fact.
Around 350 BC Eudoxus of Cnidus wrote a book, On Speeds, where he was trying to reproduce motions of the planets using combined rotations of several spheres. He noticed that the path of the planet depends not on the rotation speeds of the spheres, but only on their ratios. Riddell speculates that this work led Eudoxus to his famous theory of proportion, presented in Book V of Euclid's Elements, which became a cornerstone of Greek mathematics.