Symbolism, Imagery, Allegory
When Matt's dad gives him his precious watch, Matt knows it is a really big deal, and he handles it "as gently as if it were a bird's egg" (1.11). This is no ordinary Swatch, folks—it belonged to Matt's grandpa and is "the finest thing his father had ever possessed" (1.15). So giving it to Matt shows both the love Matt's dad has for him and his trust in his son. By placing his watch in Matt's hands, he also places his trust in Matt to care for the home and property until he returns.
So when Matt gives the watch to Attean, it's also a really big deal—when he thinks of parting with the watch, "Something twisted tight in his stomach" (22.31). But Attean has given him (1) his respect, (2) acceptance as his white brother, and (3) his prize possession, a.k.a his mangy, good-for-nothing dog.
Matt knows the watch is the best he can do to show his friendship, respect, and gratitude to Attean, and even if Attean never needs to use it, Matt knows he'll treat the watch right. For Attean, the watch will always be a reminder of Matt and their unlikely friendship.