People haven't really seemed to comment on it much, but the fact that Ludo now has a functioning wand built around the missing piece of Star's wand kind of implies the wand is able to repair itself if broken. Hence the new unicorn appearing to take the place of the one Star killed, and the broken crystal drawing other objects around it to form a second wand. Both of those occurrences imply the wand simply being broken could be reversed through its innate property, but because neither Star nor Ludo have all the pieces both are only semi-functional.
It may also be that the finger inside Star's wand was there even before the wand was destroyed, but it being cleaved was what first made its presence visible. Since of course Moon fought Toffee before Star ever had the wand, though there's always the possibility Toffee placed it in seconds before the wand exploded.
I'm not sure every detail is going to get a complete explanation at this point. Everything with the wand is oddly complex now compared to the first season.