This is a question that is difficult to answer because although the only three books referring to Michael do so in a symbolic context, that doesn’t in itself rule out that there is such an angel.
The original 3 references to Michael are in Daniel, where he is the defending chief prince of Israel, against the heavenly prince of Persia (the guardian angel of Persia?) and other nations.
The prince of the kingdom of Persia withstood me twenty-one days, but Michael, one of the chief princes, came to help me, for I was left there with the kings of Persia, (Daniel 10:13 ESV)But I will tell you what is inscribed in the book of truth: there is none who contends by my side against these except Michael, your prince. (Daniel 10:21)
“At that time shall arise Michael, the great prince who has charge of your people. And there shall be a time of trouble, such as never has been since there was a nation till that time. But at that time your people shall be delivered, everyone whose name shall be found written in the book. (Daniel12:1)
The 2 New Testament references are also in visions:
But when the archangel Michael, contending with the devil, was disputing about the body of Moses, he did not presume to pronounce a blasphemous judgment, but said, “The Lord rebuke you.”(Jude 1:9)
This is referring back to Zechariah 3:1, a parody of the corruption of the priests in the time of Zechariah and Nehemiah, although the name ‘Michael’ has been added by Jude. (For more information see a separate answer here.) Then finally we have a symbolic passage echoing Daniel 10:13, 21, 12:2 in Revelation 12:7.
Now war arose in heaven, Michael and his angels fighting against the dragon. And the dragon and his angels fought back, (Revelation12:7)
So again, there is no real evidence in the Old or New Testaments for Michael being a literal being.
Michael and Gabriel?
The only real argument for Michael being real then is by analogy with Gabriel. Because his name is linked with Gabriel (Daniel 8:16, 9:21) who appears to Daniel on earth. And of course the angel which appears to Mary in Luke 1:19, 26 identifies himself by name as Gabriel. But it could also be read that these two ‘names’ are actually titles, rather than names which a specific angel carried permanently.
Michael means “Who is like God?” – which would explain why Michael was the chief prince guardian of Israel’s army in Daniel. Gabriel means “God is my strength”, which is appropriate to his role in Daniel, though less obviously so in Luke.