Okay so I've been watching a lot of Bleach recently and I've noticed a serious trend with the battles. Before they start, both characters introduce themselves to the other. I've tried to do some research about this (mostly through Google) but I haven't found anything. Is it just something for plot convenience or is it some kind of tradition?
As mentioned above:
Its customary to announce your name before a duel because afterwards one person will be dead, preventing the victor from knowing the name of a "worthy opponent". Source - lots of manga and anime. Conversely, its highly insulting to not announce your name beforehand and shows great respect to ask the name of "worthy opponent"
This is the case in several anime dating back to before dragon ball. The reason why is because of the strong influence of honor and dignity in battle portrayed by several historical warriors like samurai (Miyamoto Musashi) and ninja (Hattori Hanzo) and even in the Japanese underground (Yakuza).
Stating one's name before entering a fight or even in the midst of a fight is greatest show of respect because whomever dies will carry the name of the one who killed them into "life-beyond-death" and the name of died was actually prevalent enough for the killer to take the time to recognize.
In some respects it even honors someone as a potential equal in terms of combat potential.