Zuko wasn't born in this time line or even demension. He was born in a different one, one where Emerald did not exist. But Zuko has a sad story, his world suffers a horrible fate. It doesn't survive. His demension itself is destroyied, and Zuko manages to get into a timemachine to get away.
However that timemachine malfunctioned, causing Zuko to land in a parrael demension not in a different time.
Alone and scared the young Zuko is unsure of what to do. Bast finds him, and immediatly notices how much he resembles her father and late grandfather. Confused, she asks him who he is and who his parents are. He tells her his name is Zuko and his parents are dead. Now very confused Bast brings Zuko back to where the Z fighters are currently holding a meeting on how to deal with the aliens attacking the young Lillia situation. Trunks and Emerald as well as everyone else are instantly hit with the similarities.
He tells his tale of woe to the sympathetic Z Fighters. Instantly the discussion begins of who will take him in. Parsley starting the case, fallowed by Kita and even Seripa. But before anyone else can pick up or the the three of them can make a decent case as to why they should be allowed to adopt Zuko, Emerald puts and end to it. She points out that not only do her and Trunks have the most space but Zuko is the son of Trunks' alternate demension counterpart. It would only be right that they take him in.
Trunks fully agrees.
He later develops a crush on than later falls in love with, Covertia. Who he affectionately calls Tia.