Luke 8:20, 21 alludes to a deeply significant concept in the theology of the Bible and especially the NT. It is the basis of several other major teachings of the NT, namely, Adoption, Reconciliation, and Prioritization. Let me take these one at a time (but only briefly).
Literal adoption was used in both Hebrew (most notably Moses, Ex 1, 2, and Esther) and Roman societies as a means of providing heirs for those who could not have their own, or, providing protection for children deprived of parents by either death or poverty. It is the latter sense that the New Testament uses the term as a metaphor of a person becoming a “Son of God”. This idea is drawn from the Old Testament that discusses Israel’s adoption by God (Deut 14:1, Jer 31:9).
The idea of sinners being adopted as Sons of God occurs infrequently in the NT and only by Paul (Rom 8:15, 23, 9:4, Gal 4:5, Eph 1:5). However, Jesus appears to unmistakably allude to adoption in John 3:1-8 and 1:12, 13 where we are able to become children of God.
The metaphor of adoption is extended by the New Testament’s repeated idea of Christ being our brother (Heb 2:11-13, Ps 22:22, Isa 8:17, 18, Matt 12:48, 49, John 20:17, Rom 8:29) following adoption.
Reconciliation describes the process of reuniting an estranged family member. It is predicated on two Biblical assumptions that (a) Jesus is our brother (Heb 2:11-13, Ps 22:22, Isa 8:17, 18, Matt 12:48, 49, John 20:17, Rom 8:29), and (b) sin separates us from Jesus our brother (Isa 59:2, Gal 5:4, Eph 2:12, Ps 22:1, Eze 14:5, Jer 6:8). Reconciliation is found in only a few places but they, again, emphasise that atonement is God’s initiative without any input from us. In 2 Cor 5:18, 19 we find that Christ reconciled the world to Himself by “not counting our sins against us”. Rom 5:10, 11 teaches that sinners were reconciled to God by Christ’s death. Further, a comparison with v9 shows that justification and reconciliation are used in parallel.
Jesus was clearly teaching in Luke 8:20, 21 that His spiritual family now took priority/precedence over his biological family. He discusses a similar idea (using Hebrew hyperbole) in other places such as, Luke 14:26 -
“If anyone comes to Me and does not hate his father and mother and
wife and children and brothers and sisters—yes, even his own life—he
cannot be My disciple."