The pertinent word in Rom 9:4 translated "adoption" is υἱοθεσία and occurs only in five places in the NT, Rom 8:15, 23, 9:4, Gal 4:5, Eph 1:5. According to BDAH it means:
adoption, a legal technical term of 'adoption' of children, in our literature, ie, Paul, only in a transferred sense of transcendent
filial relationship between God and humans (with the legal aspect, not
gender specificity, as major semantic component)
- (a) of the acceptance of the nation of Israel as sons of God (cp Ex 4:22, Isa 1:2 al., where, however, the word υἱοθεσία is lacking; it is
found nowhere in the LXX) Rom 9:4
- (b) of those who believe in Christ and are accepted by God as God's children, eg, Gal 4:5, Eph 1:5, Rom 8:15, 23 ...
The NT metaphor of adoption is used in some places as a proxy for the privileges of the covenant - a figure taken directly from the OT. That is, by being adopted in the sonship/family of God through Jesus Christ, we gain promises than cannot be alienated. See appendix below.
The metaphor of adoption is further extended in Rom 11 to that of being grafted into the "olive tree" of Israel. Paul sums up his views in Rom 9:8 -
So it is not the children of the flesh who are God’s children, but it
is the children of the promise who are regarded as offspring.
That is, Christians, followers of Christ are "Children of faith" as Paul states in Gal 3:26-29 by linking adoption with the covenant -
26 You are all sons of God through faith in Christ Jesus. 27 For all of you who were baptized into Christ have clothed yourselves
with Christ. 28 There is neither Jew nor Greek, slave nor free,
male nor female, for you are all one in Christ Jesus. 29 And if
you belong to Christ, then you are Abraham’s seed and heirs according
to the promise.
John also alludes to his spiritual adoption in the great preface to his gospel:
John 1:12, 13 - But to all who did receive Him, to those who believed
in His name, He gave the right to become children of God— children
born not of blood, nor of the desire or will of man, but born of God.
Thus, Gal 4:5 speaks of God's adoption of Israel as the chosen people and extends this same idea to the new covenant by saying that converted Christians are adopted as sons of God.
1 John 3:1 - Behold what manner of love the Father has given to us,
that we should be called children of God.
1 John 5:2 - By this we know that we love the children of God, when we
love God and obey his commandments.
APPENDIX - Adoption
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; Rom 1:7, 2 Cor 1:2, Eph 1:2, 5, Gal 1:3, 4:5, Phil 1:2, 4:20, Col 1:2, 1 Thess 3:11, 2 Thess 1:1, 2, 2:16, 1 Tim 1:2, etc. This is in contradistinction to the Jewish leaders whom Jesus accused of having the Devil as their father, John 8:44.
Thus, adoption is spoken of in the present and future tenses:
- Adoption in our current life is a metaphor of the reception of the spirit, “the spirit of adoption” (Rom 8:15) signifying a complete change of attitude and way of life which frees us from the constraints of the law, slavery to sin and fear of spiritual poverty, with the added bonus of the promise of future glory in heaven. This process is technically (theologically) called “conversion”, which see.
- Adoption in the future life (Rom 8:23) is used as a metaphor of glorification when the saints are translated to heaven.
Adoption is used as a figure of the privileges of sinners under the protection of God in the Christian life, but Gal 4:5 links the idea to redemption and hence to atonement. Thus, it is more a symbol of the change of life from sinner and assurance of heaven (that is conversion) than of only atonement. The latter (psychologically) creates the former. That is, a person of royalty is free from fear of slavery and poverty, but must be generous to those in need.
Indeed, God is frequently spoken of as the “Father” of the Israelites throughout the OT, Deut 32:6, Ps 89:26, Isa 63:8-10, 16, 64:8, Mal 1:6. In the NT writers frequently refer to God as “the Father”, Gal 1:1, Eph 1:3, 5:20, 6:23, Phil 2:11, 1 Thess 1:1, 1 Cor 15:25, 2 Cor 1:3, 11:31, James 1:27, 1 Peter 1:2, 3, 2 Peter 1:17, 2 John 3, etc; or “My Father”, Matt 11:27, 12:50, 18:35, 20:23, 26:53, Luke 10:22, 15:58, John 5:17, 8:19, 54, 10:17, 18, 29, 14:21, 23, 15:18, etc. The Lord’s Prayer begins with “Our Father”, Matt 6:9, see also Gal 1:4, 1 Thess 3:11, 2 Thess 2:16, Titus 1:4, Col 1:3, Phil 1:2, 4:20, etc.
The metaphor of adoption is used as a proxy for the promises of the covenant in Rom 9:4, Eph 1:5, Gal 3:26-29, 4:5; this is apt as the covenant and adoption grant great privileges.
The metaphor of adoption is extended by the New Testament’s repeated idea of Christ being our brother (Heb 2:11-13, 17, Ps 22:22, Isa 8:17, 18, Matt 12:48, 49, John 20:17, Rom 8:29) following adoption.