Vasa wasn't named after the royal house, it was named after the vase[0] on the heraldic symbol for the house of Vasa[1]. A vase is probably best described as a fasces. Äpplet was probably named after a globus cruciger[2], or "national apple" (riksäpple), part of the regalia of Sweden.
Gustav was known as "Gustav Eriksson" in his day, "Eriksson" being a patronym. Family names wasn't really a thing at the time, as coat of arms were. The form "Gustav Eriksson Vasa" first appears in the 17th century and simply "Gustav Vasa" isn't used before the 18th.
https://en.wikipedia.org/wiki/House_of_Vasa