Straightforwardly, Messianic Judaism is about walking in the way of the Jewish Messiah in accordance with his commands, and as such, we are bound by the Mosaic Law. It can be useful to study Rabbinic Judaism in order better understand the cultural context of the Bible, but God's Word is the highest authority, so we need to check everything they teach against it to see if it is in accordance with it. In Acts 17:11, the Bereans were praised because they diligently tested everything Paul said against OT Scriptures to see if what he said was true. The NT authors quoted or alluded to the OT thousands of times in order to establish that it supported what they said that that they did not depart from it to the right or to the left.
There are many verses that say that the Mosaic Law is for our own good to teach us how to walk in God's ways such as Deuteronomy 10:12-13, so the Law is not about how to act like a Jew, but rather it is God's instructions for how to reflect His attributes to the world, such holiness, righteousness, goodness (Romans 7:12), justice, mercy, faithfulness (Matthew 23:23), love, joy, peace, patience, kindness, gentleness, and self-control (Exodus 36:6-7, Galatians 5:21-23). So while we are not under the Mosaic Covenant, we are nevertheless till under the same God, who has the same ways, and therefore the same instructions for how to walk in His ways, and as such there is no difference between the laws of the Mosaic Covenant or the New Covenant, especially because the New Covenant involves God writing this same Law on our hearts so that we will obey it (Jeremiah 31:33).
To give another example, in 1 Peter 1:13-16, it says that we are to have a holy conduct for God is holy, which is a quote from Leviticus where God was giving instructions for how to have a holy conduct, such as Leviticus 11:44-45, so refraining from eating unclean animals is not about acting like a Jew, but rather it is about acting in accordance with the eternal holiness of our God and reflecting His holiness to the world. Jews have the the God-give role of being a light to the world, to teach the nations about God and how to walk in His ways (Isaiah 2:2-3, Isaiah 49:6, Deuteronomy 4:5-8), so the Law was given to the Jews for the purpose that they would be a blessing to the nations by teaching them how to obey it, and the reason why the Gospel always went out to the Jew first and then to the nations was so that they could fulfill this role. So again, it's not about identifying with Jews, but about identifying with God.