I like to think that the Zvahl family had come into a significant amount of emeralds generations ago, and now they pass them down as a sort of symbolic tie kind of thing. The color green, and the emerald are all symbols of the Zvahl family line. Thus why both Cadenza and Laurance are both characterized with the color green.
When Cadenza was born, she received an emerald necklace that she started wearing when she grew older, and most of her wardrobe was of various shades of green.
When Laurance was adopted, he also started receiving clothes that of various shades of green, and he received his scabbard with emeralds inlaid on its surface. In the academy he pierced his ears when he realized he wasn't standing out enough to get the attention of the jury, and when he sent a letter to Cadenza mentioning in it that he pierced his ears, she sent back two emerald studs, Laurance's second emerald heirloom.
I like to think that Lord Hayden also had a significant number of green/emerald regalia that wasn't necessarily shown in the series (minor character change).
When Katelyn gave Laurance the emerald sword, Laurance was surprised that she knew about the Zvahl family's symbol and was also shocked at the size of the emerald sword, as he's only seen emerald jewelry and bedazzlement, never a full weapon.
(You should go read my fic "Hold Tight To Me" if you like this)
