No it isn't. Without it, the meaning would be, "He is a member." What of? A choir? a church? a scout troop? a jury? an orchestra? a rock group? There are many things he could be a member of, apart from a club. You could, of course, say, "Li estas klubano," and then "de la klubo" would be redundant.
I'm not very advanced in Esperanto but my theory is that there are often subtle differences between words used as affixes, which have a grammatical funciton, and words not used as affixes which have rougly the same meaning because one has a grammatical function and one has a lexical function. "Ano," for example, means something a little more abstract and functional than "membro."
For example, you could say this: "Li simple estas membro de la asocio, sed ŝi estas la prezidanto. Kaj li kaj ŝi estas asocianoj. Ili estas anoj de la asocio"
In English, this would be a pretty close analogue: "He is simply a member of the assocation, but she is the president. Both he and she are associates. They are part of the association"
Even if the "prezidanto" of the association is not technically considered a "membro," that person could still be an "ano." And though acceptable, it's slightly incorrect to say that something like "klubano" means "club member" in the same way that it's slightly incorrect to say that "farmisto" means "farm-doer" as opposed to "farmer"
I always like to consult the PIV when it comes to the hair-splitting questions like this. The PIV directly defines "ano" as: "Membro de grupo, de societo". In contrast, "membro" appears to be for the most part, an anatomical term. It's defined directly to refer to the extremities: "Ĉiu movebla parto de la korpo, escepte de la kapo". "Membro" can be used only figuratively, to mean "Ano de societo, partio, familio, eklezio ks".