If both 五 and 伍 mean "five", when should I use each one?

伍 is an alternative Japanese numeral that is used in formal documents. This documents can include legal documents. When they are used in legal documents, these set of numerals can prevent contracts, checks, and other documents from being changed or manipulated after printing. For example, the standard kanji for 1, 2, and 3 can be incremented up by the addition of a single stroke. Therefore, more complex kanji like 壱 (one) are used to prevent this practice. We encounter the same problem with Hindu-Arabic numerals in English, so we parenthetically spell out the number in English for the same effect. For example, “the merchant agrees to deliver 2 (two) units to the customer.”

There are, however, other uses for these kanji. They have a certain cachet to them. They are old-fashioned and formal, perhaps a little more Chinese than everyday numeric kanji. In the manga you saw it in, it probably is meant to look cool. Being set in the Taisho period may have contributed to this style decision, and furthermore the “coolness factor” is applicable even to stories set in the present or future (e.g. Neon Genesis Evangelion).

伍 was used to prevent falsifying the accounts. Such older notations are called Daiji (大字) in Japanese (see Wikipedia in Japanese 大字（数字）). In formal, in Japanese law only allowed to use 壱、弐、参、拾 (one, two, three, ten) and other characters like 肆, 陸 are not used now.

I believe its a formal kanji used in legal documents. Also I read that its a jinmeiyō kanji, a kanji used for names. https://en.m.wikipedia.org/wiki/Jinmeiy%C5%8D_kanji

The author probably chose to write more formal and older kanji because the story is set in the Taisho Period. Thats just my guess.