Yes Filiberto the bird could be part of a gentleman’s, a merchant’s or a guild’s coat of arms.
Another possibility is that it was the symbolic animal of an administrative subdivision, ward or district («contrada», «rione», «sestiere» or «terziere», whatever) of an Italian city-state.
In Siena for example, each of the 59 original «contrade» was named after a symbolic animal, including several birds: Oca (goose), Gallo (rooster), Civetta (owl). Pisa and Lucca also had similar citizen organizations, if I do not err.
The tournament represented on the cassone was not necessarily celebrated in Florence, by the way.
Anyway, the odds seem to be in favor of an Italian weaving of the rug.
The golden feathers could be part of "featherman's " heaume (or great helm") which he is perhaps holding with his right hand while waiting his turn.
It will surely disappoint Steve, but I doubt that the fellow had these feathers implanted in the style of a "Moulin Rouge" dancer
Regards
Pierre