Was pi invented or discovered?

[Anomalocaris]

There is a difference between the actual speed of light in a vacuum, and c, which is an estimation of that speed. The actual speed of light is clearly a property of the physical universe; light exists and it moves. The value we have assigned c is an invented number, as accurate as we can currently get it (and it's currently based off a 2009 calculation as far as I can tell). We are trying to get more accurate, but due to the limitations you noted above, as well as computational limitations, it's unlikely we'll ever get the true value (though perhaps we may be able to infer it).

In terms of Pi, the analogy here would be that however hard you try to make a perfect circle in the physical world, if you take the circumference of that circle and divide it by the diameter, you will never end up with the actual value of Pi.

The "related properties" I assume you are talking about here are the circumference and the diameter of a perfect? If so, then what you've said is categorically wrong. Those properties do not make any "natural occurrences" in the universe, because it is impossible to find (or construct) a perfect circle in the universe. You can only get the irrational number Pi by creating a perfect circle. Any imperfect circle will get you a number that does not equal Pi (though it may be very close).

"perfectness" of the circle totally misses the point.

The understanding of a property is not the mere cataloging of its manifested impact on the measurements actually made. It includes the capacity to make predictions on what the measurement would turn out to be if the circumstances of observation were to be changed from those of the actual observation. This includes predicting behavior even in notional circumstances marginally beyond physical capacity to actually achieve. So the fact that notionally perfect circle is impossible to achieve is irrelevant. What is relevant is actual instance of roundedness can be conceptually changed to reflect this notional perfect circle and predictions can be made upon it.

So through observation of the behavior of actual ratios between measurable circumference and dimension through centroid of real and quantifiable shapes, we developed an understanding of the property of this relationship such that we could predict how the ratio would change if a theoretical arbitrary shape were to deviate from actual observed shape. We so happen to find one instance of this predicted ratio to be the most practically encapsulating and useful. This so happen to be a shape requiring a minimum of fuss to describe, and yet adaquately approximating a large and useful collection of actual applications. This we call Pi.

We didn't invent Pi. We dicovered how to predict a specific, theoretical but encapsulating instance of the manifestation of a property.