It does appear to be legitimate based on what I can see in the picture. Can observe differences in the handwriting between individuals, and also in comparing repeated characters in different signatures. IMO it’s real and would have taken too much time and knowledge/skill of Japanese writing to be faked by a GI.
I believe your flag is the real deal from WWII. First off, no GI did the writing, it is too complex and too nicely and accurately done for that to be a possibility. While there is some similarity in style in some of the writing there are also enough differences to convince me that the signatures were written by different individuals. The characters at the top of the flag, reading from right to left, say "jinchuu hokoku", a commonly encountered wartime slogan meaning roughly "loyalty and patriotism" or "be steadfastly loyal in your service to your country" or something along those lines. The characters in the meatball, reading clockwise from the upper right say "hissho-oh" meaning "must (achieve) victory" or "Gotta Win!" and "yuushi" which I think means something like "courageously do your utmost (for victory). And no, it is not uncommon to see characters written inside the meatball.