Notice that Nephi needed to say that his people should not look for another Messiah to come. There would be no other Messiah, "for there is save one Messiah spoken of by the prophets, and that Messiah is he who should be rejected of the Jews." Nephi must have been aware that some people wondered how many Messiahs there would be. What made them wonder?
The word mashiach in Hebrew means "anointed" or "anointed one." In ancient times, a high priest was anointed to be the high priest. So, in a way, every high priest under the Law of Moses was a "mashiach"—a messiah. In 1 Nephi 10:5, Lehi said that a messiah would come. He did not say the Messiah, so Nephi may have wanted to clarify this matter.
There are reasons why we have multiple anointed people, but that does not make them all the Messiah. The name "Mosiah" is actually related to the same word. "Messiah" was a word that was used in more than one way, but Nephi wanted us to know that there is only one Messiah—he who would perform the atonement and would be the one whom we worship through his holy name. As Nephi taught in verse 13: "for I have seen his day, and my heart doth magnify his holy name."
Book of Mormon Central, "How Was Mosiah a Type of Christ? (Mosiah 26:12)," KnoWhy 104 (May 20, 2016).