Anche has two meanings in Italian: perfino = even and pure = also/too. Often only the context tells us which is intended. Without context, Duo shows the first but also accepts "He too ...".
Anche relates to the preceding word only if this is a verb; in all other cases it relates to the following word. Here, that's lui.
"He too has ..." avoids the ambiguities in English of "He also has ..." and "Also he has ...". In the first case we don't know if "also" relates to "he" or the verb. In the second, your version, we don't know if you mean "he too" or you are referring back to the previous sentence with a conjuction. The best Italian word to refer back like this is Inoltre ..., and I'd choose more precise English such as "Moreover ...", "In addition ..." or such like.
This is such an awkward sentence to translate into English. "Even he has his wild side" wouldn't sound right unless the emphasis is on 'he', as in "Even HE has his wild side", meaning he has a wild side just like everyone else. I translated it as "Also, he has his wild side" meaning in addition to having a calm side, he can be wild. "He also has his wild side" is a better way of saying it, grammatically. This sentence can be translated with different meanings.
My personal interpretation (not to be confused with the CORRECT one):
He, too, has his wild side = Of all those people, he (as well) has his wild side.
He has his wild side too = He has more than one side, and one of them is wild
I think both ways can be and are used, with a slight shift of emphasis. Helpful at all?