Transcendental numbers are usually characterized by a limiting iterative process. We don't say that a sunflower has exactly the Golden Ratio's worth of seeds, either; instead, we say that the Golden Ratio is inherent in the pattern of sunflower seeds. [0]
e is most famously in compound interest, or other continuous processes. An example in nature might be the radioactive decay of certain isotopes, often indicated by half-life. In particle physics, e manifests as the limiting process of septillions of particles operating independently with each other.
pi is more abstract. It might well only show up in idealized conic sections, which mean that all the natural conic sections we see, such as shadows cast by light sources, are imprecise. (This is because, extremely technically speaking, shadows are not physical!) There are still limiting processes for pi, like taking a large chunk of matter and chilling it into a constantly-curved smooth ball, which will generate an approximate sphere, but they are much harder to perform because gravity is fickle and atom boundaries are fuzzy. (That said, we can still make such artifacts when we want to, at great cost. [1])
In both cases, our understanding of the the number's construction is good enough that we no longer need any physical props to approximate their value, but instead can compute the values abstractly and then use them to refine our physical measurements.
e is most famously in compound interest, or other continuous processes. An example in nature might be the radioactive decay of certain isotopes, often indicated by half-life. In particle physics, e manifests as the limiting process of septillions of particles operating independently with each other.
pi is more abstract. It might well only show up in idealized conic sections, which mean that all the natural conic sections we see, such as shadows cast by light sources, are imprecise. (This is because, extremely technically speaking, shadows are not physical!) There are still limiting processes for pi, like taking a large chunk of matter and chilling it into a constantly-curved smooth ball, which will generate an approximate sphere, but they are much harder to perform because gravity is fickle and atom boundaries are fuzzy. (That said, we can still make such artifacts when we want to, at great cost. [1])
In both cases, our understanding of the the number's construction is good enough that we no longer need any physical props to approximate their value, but instead can compute the values abstractly and then use them to refine our physical measurements.
[0] https://en.wikipedia.org/wiki/Spiral#In_nature
[1] https://www.youtube.com/watch?v=JKCBeDeVxkg