lol probably not, but i think its possible in the future games we might find out the origin of the crest design. or maybe its just a really old family crest going back hundreds of years. there is so much detail in shenmue that you are never sure whats significant to the story or not.
However no I don't think it does, although I wouldn't be surprised if the Hazuki family crest originated from a ancient Chinese symbol, seeing how his family seems to have a strong connection to China's past.
Perhaps this crest was given to the Hazuki family as a means for later generations to beable to identify them, as people who could be trusted or whatnot.