The definite article "the" is in parenthesis of the English translation in the Interlinear. The Greek "σαρκὶ (sarki)" is literally "in flesh". But, the uses of "sarki" in other occurrences also add the definite article "the" in the English translations. See Rom. 2:28, 7:5, 2 Cor. 10:3, Gal. 2:20.
John was denying the teaching of some of the people at that time who were falsely claiming that Christ had only been a "spirit" or some phantom appearance.
Benson's Commentary mentions the importance of "the flesh" as:
"The clause imports, 2d, That this great personage, the Messiah, the Son of God, had really come in the flesh, and had a real human nature, in opposition to a sect which arose very early in the Christian Church, called the Docetæ, who would not allow that Christ had a real body, and that he really suffered, died, and rose again. This sect St. John seems to have had in his eye throughout this epistle. Hence, in the very beginning of it, he speaks of seeing, hearing, and handling Christ; and here, to the fundamental article of Jesus’s being the Messiah, he adds, that he came in the flesh; with which doctrine his atoning for sin by the sacrifice of himself, and his rising from the dead, the first-fruits of them that sleep, were closely and necessarily connected, and therefore the acknowledgment of it was a point of the greatest importance." Source: here
Not only were the first century converts to "the way" endangering their lives by confessing Christ, but there were some sects which were denying that the Messiah actually had the body of man.. the flesh.
Excerpt from Barnes' Notes on the Bible:
"The point of the remark made by the apostle is, that the acknowledgment was to be that Christ assumed human nature; that he was really a man as he appeared to be: or that there was a real incarnation, in opposition to the opinion that he came in appearance only, or that he merely seemed to be a man, and to suffer and die. That this opinion was held by many, see the Introduction, Section III. 2. It is quite probable that the apostle here refers to such sentiments as those which were held by the "Docetae;" and that he meant to teach that it was indispensable to proper evidence that anyone came from God, that he should maintain that Jesus was truly a man, or that there was a real incarnation of the Son of God. John always regarded this as a very important point, and often refers to it, John 19:34-35; John 20:25-27; 1 John 5:6. " Same source as above.
And, Jamieson-Fausset-Brown offers a nice comment:
"Jesus Christ is come in the flesh—a twofold truth confessed, that Jesus is the Christ, and that He is come (the Greek perfect tense implies not a mere past historical fact, as the aorist would, but also the present continuance of the fact and its blessed effects) in the flesh ("clothed with flesh":" Ibid.
The definite article "the" is customary in English. We could say "in body, in mind, in spirit" or "in the body, in the mind, in the spirit" without changing the meaning.