When Jesus asks whether the Twelve also intend to leave, Peter answers that there is nowhere else to go: Jesus has the words of eternal life. He then confesses him as God's Holy One.
The title comes at the climax of a chapter filled with competing attempts to identify Jesus.
The strongest early Greek textual evidence supports 'the Holy One of God' in John 6:69, while later manuscripts include expanded readings such as 'the Christ, the Son of the living God,' probably influenced by familiar Synoptic confessions. 'Holy One' language denotes one uniquely belonging to and consecrated by God and has biblical precedents for divine holiness and chosen agents.
John's wording should therefore be preserved rather than harmonized with Matthew.
The textual history matters because John's particular confession contributes its own emphasis. Peter does not merely repeat a formula from another Gospel.
After the crowd seeks a king and many disciples reject Jesus' claims, the Twelve remain because they have come to recognize that life is found in this uniquely consecrated one sent from God.