To understand the significance of Daniel's statements regarding the captivity, Isaiah's prophecy to Hezekiah must be understood. Daniel is confirming that prophecy.
First, Daniel's statement:
3 And the king spake unto Ashpenaz the master of his
eunuchs, that he should bring certain of the children of Israel, and
of the king's seed, and of the princes; 4 Children in whom
was no blemish, but well favoured, and skilful in all wisdom, and
cunning in knowledge, and understanding science, and such as had
ability in them to stand in the king's palace, and whom they might
teach the learning and the tongue of the Chaldeans. (Daniel 1:3-4,
While Nebuchadnezzar was a wise king who respected ability and did not elevate only royalty in his courts, those of royal lineage would be in the most favorable position to fulfill the king's requirements. And his capture of the "king's seed, and of the princes" had been prophesied long beforetime.
16 And Isaiah said unto Hezekiah, Hear the word of the
LORD. 17 Behold, the days come, that all that is in thine
house, and that which thy fathers have laid up in store unto this day,
shall be carried into Babylon: nothing shall be left, saith the LORD.
18 And of thy sons that shall issue from thee, which thou
shalt beget, shall they take away; and they shall be eunuchs in the
palace of the king of Babylon. (2 Kings 20:16-18, KJV)
Daniel would certainly have been familiar with this prophecy. It is even likely that his deportment in Babylon were tempered by this prophecy. Knowing it had been predicted, and was a part of God's master plan, Daniel would have found it easier to accept the situation. Most importantly, Daniel writes in confirmation of the fulfillment of the prophecy, to strengthen the faith of those who should afterward read and understand it.
Daniel also references multiple times, as first indicated in the quote of Daniel 1:3 above, that he was among the "eunuchs." This, too, had been prophesied.
A final, minor clue, lies in the fact that "Daniel" was one of David's own sons, and the name may have been considered worthy of royalty in the generations following.
Now these were the sons of David, which were born unto him in Hebron;
the firstborn Amnon, of Ahinoam the Jezreelitess; the second Daniel,
of Abigail the Carmelitess: (1 Chronicles 3:1, KJV)
While we do not have full confirmation (proof) of Daniel's lineage, we have sufficient evidence to assert with confidence that Daniel was indeed of the royal line of Judah, even as rabbinic tradition claims. His own testimony confirming the fulfillment of Isaiah's prophecy gives great weight to this conclusion.