Please allow me first to remind the readers that the emotive meaning applied to TISHA BE'AV (the ninth day of the Jewish month of Av) includes not only the destruction of the First and Second Temples, but several other catastrophes, such as the Expulsion from Spain.
And the main point is that in Hebrew, 'First Temple' refers to 'BAYIT RISHON,' and 'Second Temple' refers to 'BAYIT SHENI,' they being the FIRST and SECOND COMMONWEALTHS, each one meaning the Sovereign LAND OF ISRAEL (ERETZ ISRAEL).
This First Bayit was ended ('destroyed') when Nebuchadnezzar's armies overpowered the Kingdom of Judea (Yehuda), the remaining part of Eretz Israel (after the northern Kingdom of Israel had been utterly extinguished by the Assyrians. Solomon's temple was erased (destroyed). .
Yet the Second Commonwealth, or Second Sovereign Land of Israel, Bayit Sheni --wrongly translated "Second Temple"-- started when the Hasmonean family (the 'Maccabeans,' or 'Maccabees') lead the Judeans ('Jews') to triumph against the Hellenistic Syrian armies of Antiochus IV.
Since 1948 we live in our Third Commonwealth, our BAYIT SHLISHI. And I feel proud of having volunteered from overseas to help make it a reality. But we cannot yet build a Third Temple of Worship on top of Mount Moriah, in our Capital City, Jerusalem...
The first temple (a building devoted to God's worship) was built by Solomon, and the second temple by Zerubbabel, both on top of Mount Moriah in Jerusalem, the city that David conquered and established as Eretz Israel's capital. The second temple was rebuilt and embellished by Herod. It had not been completed when destroyed by Titus in the year 70.
Yet the First Commonwealth (Bayit Rishon) started from the beginning of the Land of Israel's own sovereign government, starting with the invasion of Canaan by the Israelite tribes under Joshua, but mainly with the Judges and King Saul, consolidating definitely under King David.
Then Cyrus, the mighty Persian, conquered Babylon, restored rights to the Jews, and allowed the reconstruction of another --Zerubbabel's-- temple, later-on known as Herod's temple.
The cleansing and 'Rededication' --"Hanukah"-- of the one and only Jewish Worship Temple, signals the beginning of the Second Commonwealth. Which was ended when the mighty Romans were invited to take possession of it by stupid Jewish rulers. The second temple (Zerubbabel-Herod's) was destroyed much later, being it just a building, not the essence of a Free People, the People of the Sabbath. Indeed, it was only when the Temple on Mount Moriah stood on sovereign territory, that it merited being also referred to as BAYIT, First and Second...
So, do you now understand why Jews mourn on Tishah Be'Av, and jubilate on Hanukah?