Yes, the are interchangeable. Young's Literal Translation has either "the day of our Lord Jesus Christ", or "the day of our Lord Jesus", or as in 2 Thess. 2:2, "the day of Christ". The meaning is the same meaning as was used throughout the OT for "the day of the Lord." It was a day of judgment of wicked people, or a particular country / nation who would not repent.
Isa. 2:12,
" For the day of the Lord of hosts shall be upon every one that is proud and lofty, and upon every one that is lifted up; and he shall be brought low:" (KJV)
Isa. 13:9,
" Behold, the day of the Lord cometh, cruel both with wrath and fierce anger, to lay the land desolate: and he shall destroy the sinners thereof out of it." (KJV)
Jer. 46:10,
"For this is the day of the Lord God of hosts, a day of vengeance, that he may avenge him of his adversaries: and the sword shall devour, and it shall be satiate and made drunk with their blood: for the Lord God of hosts hath a sacrifice in the north country by the river Euphrates." (KJV)
Ezek. 13:5,
"Ye have not gone up into the gaps, neither made up the hedge for the house of Israel to stand in the battle in the day of the Lord." (KJV)
Amos 5:18,
"Woe unto you that desire the day of the Lord! to what end is it for you? the day of the Lord is darkness, and not light." (KJV)
And, the phrases that are synonymous with "the day of the Lord" are "the day of visitation" (Isa. 10:3), the "time of their visitation" (Jer. 10:1), "the year of their visitation" (Jer. 48:44), "day of wrath" (Job 21:30; Zeph. 1:15; Rom. 2:5), and "day(s) of vengeance" (Isa. 61:2; 63:4; Jer. 46:10; Luke 21:22).
All of these are associated with the anger and wrath of God. Just so, when used again in the NT it spoke of the wrath of God, and the judgment that was coming upon them for crucifying Christ and persecuting His church.
2 Pet. 3:10,
"But the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are therein shall be burned up." (KJV)
2 Pet. 3:10 is the same judgment prophesied from Ezek 22:19-22:
"Therefore thus saith the Lord God; Because ye are all become dross, behold, therefore I will gather you into the midst of Jerusalem.
20 As they gather silver, and brass, and iron, and lead, and tin, into the midst of the furnace, to blow the fire upon it, to melt it; so will I gather you in mine anger and in my fury, and I will leave you there, and melt you.
21 Yea, I will gather you, and blow upon you in the fire of my wrath, and ye shall be melted in the midst therof.
22 As silver is melted in the midst of the furnace, so shall ye be melted in the midst thereof; and ye shall know that I the Lord have poured out my fury upon you." (KJV)
This was judgment language which we should all be familiar with from the OT, and spoke of the "heavens" that would pass. This was not speaking of the heaven where God sits on His throne as that will never pass away. So, then the "heavens" that would pass were the ruling authorities of the nation or nations under judgment. See Isa. 13:13; 34:4; 51:6; Joel 2:10; 3:16; Hag. 2:6, etc.
The judgment day spoken of throughout the NT was the judgment that was coming upon Jerusalem for their rejection of the Messiah. Paul was comforting the Thessalonians who were asking if Christ had come yet. Read 2 Thess. 2:2-5 carefully,
"That ye be not soon shaken in mind, or be troubled, neither by spirit, nor by word, nor by letter as from us, as that the day of Christ is at hand.
3 Let no man deceive you by any means: for that day shall not come, except there come a falling away first, and that man of sin be revealed, the son of perdition;
4 Who opposeth and exalteth himself above all that is called God, or that is worshipped; so that he as God sitteth in the temple of God, shewing himself that he is God.
5 Remember ye not, that, when I was yet with you, I told you these things?" (KJV)
Paul was reminding them of what to look for. If they had to ask Paul if the day of Christ had already happened, then they were afraid they had missed it. If "the day of Christ" meant "the end of the world" which so many today are making it to be, then why would the Thessalonians be wondering if they had missed the "end of the world"?
A day of judgment came upon wicked people after they had been warned of His coming, of His visitation, of His wrath, of His anger. Christ's coming was a day of judgment against the tribes of Israel and they saw it by what had been foretold in Matt. 24, Luke 21, Mark 13... the destruction of that temple.
They were to watch for certain signs that would happen, and one of those was the lawless one, the man of sin. And, as Christ was speaking to His disciples - Jews - the warning was to Israel, and the land under judgment was Judea and the surrounding areas of Palestine. Therefore, the man of sin would have been someone directly involved with deceiving those in Judea and Jerusalem.
The lawlessness of which Paul spoke was already at work while he spoke to them. Therefore it was someone who was already living and was about to be revealed to them. Someone in the first century A.D. was the lawless one.
Many throughout history have associated this man of sin with Nero.
"“And after many plagues completed in the world, in the end he says that a beast ascended from the abyss…that is, of the Romans. Moreover that he was in the kingdom of the Romans, and that he was among the Caesars. The Apostle Paul also bears witness, for he says to the Thessalonians: Let him who now restraineth restrain, until he be taken out of the way; and then shall appear the Wicked One, even he whose coming is after the working of Satan, with signs an lying wonders.’ And that they might know that he should come who then was the prince, he adds: ‘He already endeavours after the secret of mischief’ – that is, the mischief which he is about to do he strives to do secretly; but he is not raised up by his own power, nor by that of his father, but by command of God.” (From Victorinus, Commentary on the Apocalypse, ad 11:7; Ante-Nicene Fathers, p. 354.)
Others believe that the man of sin was one of the Zealots that caused the riots in Jerusalem, possibly John of Gischala:
"This John physically entered the Temple, presented himself to the Zealots as a God-sent ambassador; and persuaded them to defy the laws of Rome and go to war to gain independence. He also instigated the calling in the Idumaeans to keep the Jewish sympathizers from submitting to Rome. He ordered the death of Ananus and the removal of the priesthood. After these atrocities, he became the official leader of the Zealot group m control of theTemple area-john held the temple” and began disregarded the laws of Rome, God, and man, and promising deliverance from the Romans. " From John Noe's post "What About Paul's Man of Sin" here.
The day of Christ, the day of our Lord Jesus Christ, and the day of Jesus Christ throughout the NT was the judgment day that was "at hand" when the books were written in the first century A.D. and was seen - observed - in the destruction of Jerusalem and the temple. The abomination of desolation - that which caused the desolation - was the Roman army whom God used to retaliate against the Jews.
They had used the Romans to carry out the crucifixion of our Lord and Savior. God then used the Romans to deliver His judgment in the day of His wrath in A.D. 70.
Further reading:
Exposition of II Thessalonians 1 & 2 here
My post "Days of His Visitation" at my blog ShreddingTheVeil
(All bold emphasis is mine.)