I think his goal is the same in general as others. "I want to become THE BEST!!!!"

Did Naruto at the start of the Naruto original series really want to become Hokage, or did he just want people to notice him and respect him? He certainly was in it for different reasons at the end of Shippuden, when he saw that being Hokage was primarily soul-crushing paperwork.

I think Asta at least starts with this same general premise. He is the only one without magic so he feels inferior to everybody and wants to be Wizard King so people don't see him as inferior anymore. I think him wanting to marry Sister Lily is something similar. In Asta's case, it is also true that he is extremely dumb despite being powerful. He's got princesses wanting to have his babies and keeps talking about Sister Lily, probably because his last two brain cells were beaten out of him during the last mission.