First of all, only Hashirama was able to use the Wood Style, elemental Kekkei Genkai's are something you have to be born with and otherwise cannot do. Naruto can probably use Wood-Style now due to Hashirama's cells, but I doubt he can use the 1000 Hands technique.
Asura's ability to use the Wood Style is simply filler and nothing more, in the Manga he was able to use RSM and control the Kyubi's chakra.
Asura could not and cannot use Wood Style, neither can Naruto. If Asura's reincarnates can use Wood-Style, then we would have to assume that anyone that is born with Indra's chakra will inherit the Sharingan, despite both Kekkei Genkai's being on the verge of utmost disappearance.