In my opinion, the poem does nothing to enhance or detract the possibility of the flag being authentic. On the surface it seems slightly unusual to have such a large space on the flag dominated by a poem that most people are unable to decipher. It is like signing someone's yearbook with a line from Beowolf written in the original Saxon. Next to the poem is the near ubiquitous good luck in battle phrase: 祈武運長久. Its written with a line break in the middle, which is also slightly unusual. Normally you find this phrase written prominently, without a line break. But here too, its just unusual, and not really a clue as to its authenticity or inauthenticity. The names are normal names. The recipient's name (Tsutake, Tsuzuku) is a somewhat rare last name. If you have too many of these "slight rarities" popping up, one starts to question the authenticity, but that's about all I can say.
What lends authenticity to these kinds of flags are specific names of factories or schools or clubs that present the flags, or place names that tie a flag to a real location in Japan. And of course if the flag has a date on it, and has the other hallmarks of authenticity, one's confidence increases. This particular flag has none of those hallmarks, but is it a fabrication? Hard to say.