And Josephus referred to 12 other people with the name Jesus.
Which brings us to an important point: names and identification. Here's a world in which so many people have the same name (there weren't nearly as many given names/first names as there are today) and no last names to speak of. How did people keep straight who they were writing or talking about? Through identification formulae. A common method was genitive kinship constructions. Another was nicknames, adjectives, titles, etc. Josephus refers to Jesus using the following: τὸν ἀδελφὸν Ἰησοῦ τοῦ λεγομένου Χριστοῦ, Ἰάκωβος ὄνομα αὐτῷ/"the brother of Jesus, (the one called Christ), James by name." Here Josephus seek to identify which James he is talking about; to ensure that this James is not confused with a plethora of others by the same name. So he identifies James using the genitive of kinship: this James is the brother of Jesus. But which Jesus? Josephus ensures that this is clear as well by saying it is not e.g., Jesus son of Damneus or any number of others, but rather the one called Christ.
He doesn't state that Jesus
is the Christ/Messiah. He simply uses the fact that this is a unique enough title/identifier to ensure his readers know which Jesus he is talking about.
Imagine the confusion when people are talking about Jesus here and there, Messiahs here and there, Christs (Messiah) here and there, dying, revolting, talking, preaching, and making noise, and then try to figure out what who did when and how.
1) The potential confusion of having many people with the same name wasn't some baffling thing nobody got. Just like today, authors had at their disposal multiple ways to indicate who they were talking about, and did so.
2) People weren't really talking about Christs here and there "dying, revolting, talking, preaching" etc. There were a few messianic claimants, and numerous teachers/leaders who were not, but this doesn't change the documentation we have. People in the first century weren't idiots. They didn't suddenly invent historical fiction and then forget that it was fiction.