Does unity drag coefficient have special meaning?

Does a drag coefficient of exactly one correspond to any significant shape?

Cd = 1 and Cd = F / (1/2 * A * v^2 * ro) <=> F = (1/2) A * v^2 * ro That can occur with different combinations of: fluid, area, speed and drag. So it does not imply a particular shape.

Is Cd a result of the arbitrary choice of the definitions of the SI units?

Note that Cd has no units, therefore the value will not change provided that the system of units used is consistent (SI is a consistent system of units, but there are others). Note that the expression is non dimensional:

[] means dimensions, [l] are dimensions of length, [t] dimensions of time, [m] dimensions of mass and so on.

From the expression of Cd we have [Cd] = [F] / (1/2 * [A] * [v]^2 * [ro]) = [l][t]^-2 / ([l]^2 ([l].[t]^-1)^2 . [m].[l]^-3) <=> (after simplification) [Cd] = 1 (no units)

Conclusion: Cd depends only on the shape of the object and type of surface. Just like as it happens with the phenomena of friction, there is only need for a single experimental constant. However Cd = 1 has no particular special meaning in my view.

Recommendation: It might be worth to have a look on the so called "Buckingham Pi theorem" and the subject of dimensional analysis, for a deeper compression of this and other experimental phenomenas and constants.

Good luck for your studies.