As noted in @MikeSangrey’s comment, the KJV's rendering is likely a function of the Textus Receptus, the text of Mt 16:13 of which has the personal pronoun με (Strong’s G3165 meaning me:—I, me, my).
Matthew 16:13b - Reverse interlinear (blueletterbible)
Τίνα ἄνθρωποι λέγουσιν με τὸν υἱὸν ἀνθρώπου εἰναι
Whom do men say that I the Son of Man am?
(Note that the verb εἰναι in Mt 16:13 (Strong’s G2511) is actually in the form of the infinitive and therefore has neither person nor number. For an English translation that preserves the infinitive form of εἰναι, see YLT. YLT is also worth noting because it offers yet another perspective on Mt 16:13.)
While I cannot speak to the accuracy of the Textus Receptus, nor that of the KJV translation, I do think the difference between the KJV and the other English translations has a significant impact on meaning. Regardless of the translation, there is little doubt that "the Son of Man" refers to Jesus. That said, the use of different grammatical persons effectively alters the relationship between Jesus and the title “Son of Man.”
The OP states, “The K.J.V. seems, to me, to have Jesus owning personally that he is the Son of Man.” I agree. The use of the first person is the equivalent of Jesus saying, “I am the Son of Man.” The effect of the third person, on the other hand, is to turn “Son of Man” into an impersonal title. In other words, the use of the third person creates conceptual distance between Jesus and the title “Son of Man.”
In context Jesus asked his disciples two questions about who people say he is: one from the point of view of people in general and the second from the perspective of his disciples in particular.
Matthew 16:13-16 ESV (break added after v14 )
13 he asked his disciples, “Who do people say that the Son
of Man is?” 14 And they said, “Some say John the Baptist,
others say Elijah, and others Jeremiah or one of the prophets.”
15 He said to them, “But who do you say that I am?”
16 Simon Peter replied, “You are the Christ, the Son of the
living God.”
In the course of their conversation, two identities come into view: Son of Man and Son of God. When the first person is used in both questions, the parallelism creates a sense of parity between these two identities.
KJV (select text from Mt 16:13-16)
Whom do men say that I the Son of Man am?
But whom say ye that I am? And Simon Peter answered and said, the Son
of the living God.
In translations that shift from the third person in the first question to the first person in the second, an idiosyncrasy is created that serves to accentuate the differences between the two questions and the corresponding identities. In the third person, “Son of Man” becomes more of an impersonal persona, an outer or assumed aspect of Jesus’ person, not his true self.
ESV (select text from Mt 16:13-16)
“Who do people say that the Son of Man is”?
“But who do you say that I am?” Simon Peter replied, “You are the
Christ, the Son of the living God.”
In summary, the pairing of the first person with “Son of God” is indicative of a closer connection than that of the third person and “Son of Man.” Based on this discussion, I understand the closer connection to imply that, of the two titles, “Son of God” is more expressive of who Jesus truly is (cf Php 2:6-8, Rom 8:3, Jn 1:14).