How to get my own duck for the duck race?
Prerequisites:
- you’re on disc 2 (for the Dreamcast version)
- you’ve catched the falling leaves for the first time (when trying to read the Wulinshu)
- you’ve not started looking for Ren yet
- you’ve unlocked the Duck Race
(tl;dr: Gameplay video. From 2:37 to 4:27, you can see how Ryo gets the duck, and from 4:28 to 6:48, you can see how Ryo enters the race with his own duck and wins the Big Gold Medal.)
Then:
Visit the Duck Race.
Place a bet on a duck. (I’m not sure if this duck has to come in in first place.)
Visit the backyard of the Man Mo Temple (Wan Chai → Wise Men’s Qr. → Scarlet Hills).
Go to the tree.
Catch two (or more) leaves at once.
Catch another two (or more) leaves at once.
(You can hear the duck quacking.) Now a white duck feather falls down. Catch it.
(You can see the duck sitting in the tree.) Select "Catch".
(You put the duck in your inventory.)
When visiting the Duck Race, select "Entry" to enter the race with your own duck (which you have to control).
If you win, you get the Big Gold Medal. Your duck’s skills will improve the each race (I guess it’s impossible to win without certain skill levels). After a few races (7 or so) your duck will fly down the stairs.
Trivia:
Your duck speaks, so you might want to turn on subtitles.
Usually, you’ll race against white ducks, but you might also race against a mallard and its ducklings (like in the video linked above; they probably appear if your duck gets better; and the mallard is probably faster than the usual white ducks), and on a rainy day, you might race against a penguin.
References: Duck Racing: Obtaining your own racing duck (Neoseeker Shenmue Wiki) · Any tips on winning the duck races? (GameFAQs thread) · Shenmue II Secrets FAQ for Dreamcast (GameFAQs)