The rabbinic scheme is symmetrical and satisfying: Leviathan holds the sea, Behemoth the land, and the Ziz the air. That symmetry is probably the reason the Ziz exists at all. The other two are described at length in scripture. The Ziz is not, and the verse usually offered for it, in the fiftieth Psalm, contains the phrase ziz sadai, which most translators render as something like all that moves in the field rather than as the name of a bird.
Once the name was fixed, description followed. The Talmudic and midrashic material gives the Ziz a wingspan that blocks the sun and legs that reach the seabed while its head is in the clouds, and it joins the other two in the banquet prepared for the righteous. The three are then read as a set covering the created orders, which is a theological argument in the form of a menagerie.
The Ziz is worth keeping precisely because its foundations are visible. Most legendary creatures arrive with their origins already lost. Here the whole process is on the record: an ambiguous phrase, a reading that turns it into a name, a symmetry that demands a third member, and a body supplied to fill the role. Illuminated manuscripts later drew it, and by then it was an animal like any other.