The striking thing about this line in the Gospels isn't that it's about taxation in general; it's that it's about Roman taxation.
But first some background...
By the first century CE, the Jewish people had a long history of being ruled by foreign empires. The Jews came under Babylonian rule in 586 BCE, then Persian rule in 539. Once Alexander the Great swept through the region, and his empire was split up upon his death in 323, Jerusalem found itself under the Seleucid Empire.
Seleucid rule was a particularly troubled period in Jewish history because of Antiochus IV Epiphanes, who outlawed Jewish religious practices during his reign (175–164 BCE). The Jewish memory of his reign is found in 1 Maccabees, a text written around the late second century BCE. 1 Maccabees 1:60–61 can give you some idea of how bad things were for Jews:
According to the decree, they put to death the women who had their children circumcised, and their families and those who circumcised them; and they hung the infants from their mothers' necks.
In response to persecution under Antiochus IV, Judas Maccabeus led a revolt against Seleucid rule, resulting in Jewish self-rule for the first time since the sixth century. But it didn't last long. They were only independent for less than a century until Pompey took Jerusalem under Roman rule in 63 BCE.
Given this history, it's hard to overstate just how much some Jews hated being ruled by foreign empires. To give one example, the Psalms of Solomon, written in the late first century BCE, carry a characteristic disgust for Roman rule; Psalm 2:22–35, for instance, rejoices at Pompey's death, and psalm 17 anxiously looks forward to renewed Jewish self-rule.
Given that history, and given the natural human tendency to hate taxes, it's hardly a surprise that there was a revolt. Judas of Galilee (not to be confused with the disciple of Jesus) sought to resist a Roman census for tax purposes around 7 CE. As Josephus tells us in book 18 of his Antiquities,
Moreover, Cyrenius [a Roman senator] came himself into Judea, which was now added to the province of Syria, to take an account of their substance, and to dispose of Archelaus's money; but the Jews, although at the beginning they took the report of a taxation heinously, yet did they leave off any further opposition to it, by the persuasion of Joazar, who was the son of Beethus, and high priest; so they, being over-pesuaded by Joazar's words, gave an account of their estates, without any dispute about it. Yet was there one Judas, a Gaulonite, of a city whose name was Gamala, who, taking with him Sadduc, a Pharisee, became zealous to draw them to a revolt, who both said that this taxation was no better than an introduction to slavery, and exhorted the nation to assert their liberty
Josephus tells us that this Judas began the Jewish sect called the Zealots, who sought to restore Jewish self-rule by force. He blames this sect for the Jewish war against Roman rule in the late 60s CE that resulted in the destruction of Jerusalem.
All this is very much a live issue as Jesus purportedly said to render to Caesar what is Caesar's. You ask about the "accepted Jewish teaching." That's a very hard thing to pin down on most issues in first century Judaism; there's a reason that it's something of a trope to talk about Judaisms in this period rather than a singular Judaism. The Roman taxes in particular were a contentious issue. The Jews who had a relatively warm relationship with the Romans (e.g. Herod, the Jewish king installed by Rome; the priests), were comfortable with Roman taxes; in the passage quoted above, Josephus tells us that it was a high priest who pacified Jews who were angry about this tax. But other Jews, like the Pharisees and Zealots, were quite displeased about Roman taxes; there's a reason that the Gospels depict Pharisees' disgust with tax-collectors for the Romans (e.g. Luke 5:27–30).
This pressure-cooker situation makes Jesus' statement all the more striking. Give to Caesar what is Caesar's, he says, pacifying the Herodians. But give to God what is God's. If Caesar is to be given that which bears Caesar's image (i.e. coins), what is to be given to God? What bears God's image? What a clever text.
This being my first comment on /r/AskHistorians, I may be wrong, but I don't think I need to footnote every assertion I make. As I typed out this comment, I referenced the following (in addition to the primary sources I mentioned above):
The HarperCollins Study Bible, edited by Attridge and Meeks
The Jewish Study Bible, 2nd ed., edited by Adele Berlin and Marc Zvi Brettler
John J. Collins, The Scepter and the Star: Messianism in Light of the Dead Sea Scrolls.