Discovery: Menaechmus is credited with the discovery of conic sections around the years 360-350 B.C.; he used them to solve the problem of "doubling the cube". The construction required a parabola, which he called "a section of a right-angled cone", and a hyperbola, "a section of an obtuse-angled cone". (The names parabola and hyperbola themselves are due to Apollonius.) 
Origin: Euclid notes in his Phaenomena that a cylinder cut by a plane not parallel to the base results in a section which resembles a "shield". Eratosthenes implied that Menaechmus arrived at his sections by cutting a cone "in triads".
In connection with the suggestion of the OP that "the shadow of a coin" might have led to an early observation of conic sections: It is quite possible that the ancient Greeks would have been aware of conic sections when constructing sundials, since a sheaf of light rays is a cone which is cut by the plane of the horizon in a hyperbola, and a portion of that hyperbola is then marked out on the sundial .
One may speculate about the circumstances that might have led
Menaechmus to discover the curves. Conceivably he could have developed
the idea from observing a volcano, or an anthill scuffed off by his
sandal, or some artefact like a sharpened stake. However, there is
much to recommend the conjecture [going back to Philippe de la Hire,
1682] that the sundial was the most probable basis for discovery of
the conics: A dial traced on an appropriate oblique plane could show
all three conics at any latitude.
 Conic Sections in Ancient Greece, K. Schmarge (1999).
 Early sundials and the discovery of conic sections, W.W. Dolan (1972).