... I mean, the actual statuette that was used in the motion picture. And what
was it actually made of? Does anyone know?

I found this info on the site moderntimes.com:

Seven plaster figurines of the falcon were used during the 1941 production and later displayed at the Los Angeles County Museum of Art, from which one of them was stolen in 1974, during the production of "The Black Bird" [a spoof starring George Segal]. It was alleged that the “disappearance” of the figurine and the Segal film were designed to work in tandem, that the missing falcon served to hype the new picture, and that it was all a publicity stunt. If it was, it backfired, since news accounts of the missing falcon exceeded those of the anemic Segal film.