It is Kim's favorite store at Middleton Mall, and frequently shops here, although she is also known to (reluctantly) shop at Smarty Mart. Her friend Monique works at Club Banana and they first met there.
While Kim and Monique are Club Banana enthusiasts, their friend, Ron Stoppable, is a Smarty Mart fan and shops there frequently. Club Banana & Smarty Mart are owned by the same corporation, and there is apparently no actual difference (aside from price, names of colors, and a Club Banana logo) in the products the two stores sell.
In the alternate future of “A Sitch in Time”, Shego renames Club Banana as "Club Shego", replacing all the clothing with green and black outfits derived from her own costume.
As of season four, Monique got Kim a job at Club Banana as a sales assistant, while Ron works at Smarty Mart.
In "Trading Faces", it is revealed there's a Country Club Banana located in the world's richest mall in Beverly Hills. The store is so expensive you need to pass a credit check before entry is allowed. Since Smarty Mart and Club Banana are owned by the same corporation, and Martin Smarty founded and still owns the former, it's reasonable to assume he also owns Club Banana.
At one point, Club Banana features Kim's personal mission wear; this becomes a very popular clothing trend among Kim's peers. ("Kimitation Nation")
Club Banana is probably a spoof on the clothing stores "Banana Republic," and "Club Monaco", also trendy, upscale stores.