The era that correlates to the Islamic "Golden Age" was a pretty important one in terms of interreligious relations across Europe, the Mediterranean, and the wider Muslim world. While most Christian scholars outside of the Arabic world did not have much Quranic proficiency until late in the Middle Ages (as late as the twelfth of thirteen even in the Greek world, according to R. Y. Ebied and David Thomas), criticism of Christianity and Judaism was baked into the foundations of Islam.

Depictions of Christian and Jewish stories appear in the Quran and other early Islamic literatures. You might have heard that Mary is the woman mentioned most frequently in its text-- the Quran largely agrees with Christian depictions of the life of Jesus, although it doesn't agree with the orthodox Christian belief in Jesus's divinity. From the earliest period there seems to at least be an understanding of Jewish and Christian narratives among the religion-crafters, if not necessarily familiarity with the texts itself (although that still existed primarily among scholars).

In other cases, Muslim scholars argued that Jewish and Christian scriptures had been falsified to remove accurate depiction of the "true" religion, which they claimed was the Islam that Muhammad delivered. This is known as tahrif.

Two particular targets of tahrif accusations are Ezra, a Jewish scribe whose Books in the Bible depict the end of the Exile and the construction of the Second Temple, and Paul, whose interpretation of Christianity is now the dominant one. Both were accused of altering the correct narrative to introduce blasphemous or heretical beliefs. Ibn Hazm was one of the major scholars of this theory, who argued that verses like Genesis 3:22 ("This man has become like one of us") are evidence of corruption in favour of polytheism. These verses have other interpretations among other groups, such as the Royal We, a heavenly council comprised of God and the angels, the Trinity, etc., etc., but the simplest explanation might be that the original text was composed in a time when outright monotheism (as opposed to preferential worship, called henotheism) was either developing or had not yet entered into the Hebrew religion. Paul is accused of having deified Jesus and that his interpretations of the religion were the foundation for modern Christian mistakes. Frankly, these accusations in particular are probably mostly true-- Paul's letters form a disproportionate amount of the Christian canon and are a significant part of the foundations for the religion's day to day practices.

Having developed after both Judaism and Christianity, Islam benefited from their existing history of criticism between themselves. Jewish accusations of Christian polytheism through the Trinity and Christian accusations that Jews had killed Jesus, who Muslims see as the legitimate Messiah, were echoed by Muslims.

Another way of gaining insight into its predecessor religion was through conversion. Converts play a disproportionately large role in the field of polemics (interreligious criticism). These converts specifically tended to be upper-class or well-educated members of their original religion who came into contact with other religions through study or cosmopolitan relations. For Islam, particular examples include Samaw'al al-Maghribi, a former Jew who converted to Islam, and Abd Allah al-Turjuman, a former Christian priest. These converts were able to bring specific insights into their original religion and criticize them as if from within. One writer, Abd al-Haqq al-Islami, agreed with the Muslim claim that Jewish scriptures had been falsified.

References:

R. Y. Ebied, David Thomas, and Shams al-Dīn Muḥammad ibn Abī Ṭālib Dimashqī, Muslim-Christian Polemic during the Crusades: The Letter from the People of Cyprus and Ibn Abī Ṭālib Al-Dimashqī’s Response, The History of Christian-Muslim Relations, v. 2 (Leiden ; Boston: Brill, 2005).

Berman, Joshua. "The Biblical Criticism of Ibn Hazm the Andalusian: A Medieval Control for Modern Diachronic Method." Journal of Biblical Literature 138, no. 2 (2019).