No, He Used it in Context
Paul appears to be citing, not Dt 17:6, but Dt 19:15, which more generally relates to any sin (all Scripture quotes NKJV; emphasis added):
One witness shall not rise against a man concerning any iniquity or
any sin that he commits; by the mouth of two or three witnesses the
matter shall be established.
Your belief is correct:
I was led to believe that the two or three witnesses in Deuteronomy
had to be different people, not the same person verifying something
two or three times.
Paul is giving the warning in the context of his coming the third time, not because he is going to be a witness a third time himself. Rather, he is responding to testimony from two others, Titus and the other brother that went with him to Corinth about the offering to the churches (2 Cor 8:16-24). They had returned with some good news, e.g. 2 Cor 7:13:
13 Therefore we have been comforted in your comfort. And we rejoiced
exceedingly more for the joy of Titus, because his spirit has been
refreshed by you all.
But also apparently some bad news, that some were accusing Paul and them of. Specifically those in 2 Cor 10:2b:
who think of us as if we walked according to the flesh
which accusation was related to thinking they were cheating the church of Corinth (2 Cor 7:2):
Open your hearts to us. We have wronged no one, we have corrupted no
one, we have cheated no one.
Specifically, it seems, a charge that they were seeking monetary funds only for themselves to fulfill their own earthly desires. Paul defends himself in chapter 11, feeling like a fool in doing so (e.g. 11:1), but specifically noting in 2 Cor 11:7-9:
7 Did I commit sin in humbling myself that you might be exalted,
because I preached the gospel of God to you free of charge? 8 I robbed
other churches, taking wages from them to minister to you. 9 And when
I was present with you, and in need, I was a burden to no one, for
what I lacked the brethren who came from Macedonia supplied. And in
everything I kept myself from being burdensome to you, and so I will
His defense includes a promise to continue to not take funds, which is unlike those that are making the charges themselves (2 Cor 11:12-13, emphasis added):
12 But what I do, I will also continue to do, that I may cut off
the opportunity from those who desire an opportunity to be regarded
just as we are in the things of which they boast. 13 For such are
false apostles, deceitful workers, transforming themselves into
apostles of Christ.
Paul continues to indicate that he had every right in his authority to be funded by the church, but did not take such. And further, neither did Titus or the other brother take for themselves, for Paul states in 12:16-18 about this charge:
16 But be that as it may, I did not burden you. Nevertheless, being
crafty, I caught you by cunning! 17 Did I take advantage of you by any
of those whom I sent to you? 18 I urged Titus, and sent our brother
with him. Did Titus take advantage of you? Did we not walk in the same
spirit? Did we not walk in the same steps?
So Paul is urging those making the accusations to repentance from such charges, so that when he does come the third time (2 Cor 12:14), he will not become a third witness (along with Titus and this brother) of their sins in this matter (2 Cor 12:20-21, emphasis added):
20 For I fear lest, when I come, I shall not find you such as I wish,
and that I shall be found by you such as you do not wish; lest there
be contentions, jealousies, outbursts of wrath, selfish ambitions,
backbitings, whisperings, conceits, tumults; 21 lest, when I come
again, my God will humble me among you, and I shall mourn for many who
have sinned before and have not repented of the uncleanness,
fornication, and lewdness which they have practiced.
Titus and the brother that accompanied him to Corinth for the offering had brought back good news and bad news of matters. Paul was counting these two witnesses as enough to rebuke the church in the letter here, but also was urging for those so falsely charging to repent. When he was to come, a third time to Corinth, if they were to continue in the charge, he would became a third witness against them, and he was not going to spare his displeasure based on these three different witnesses against them (2 Cor 13:2).
The fact that it would be his third time to Corinth was incidental to the fact that he was citing the passage that also used the number three as a basis of witness for one to be found guilty, but it appears he was using the fact that he had given warning the second trip, and would be coming the third, to emphasize the point. That is, in 2 Cor 13:1-2 he is using these incidental similarities together in his letter to emphasize both the fact that he is coming again and the fact that in doing so, he will be a third witness against them if they do not repent.