Ephesians 1:5 says God "predestinated us unto the adoption of children by Jesus Christ to himself, according to the good pleasure of his will." The word translated "adoption" — huiothesia — is a specific legal term in Roman law: the formal act of taking a child not born to you and establishing them with full legal rights as your own. Paul chose this legal term deliberately. God's adoption of believers is not metaphorical generosity — it is a legally precise, binding commitment with full rights and standing. What adoptive parents do in courtrooms is a reflection of this.
Isaiah 43:1 — "I have called thee by thy name; thou art mine" — is God speaking to Israel, a people he formed and claimed. The claiming of a specific name, followed by the possessive "thou art mine," is the structure of adoption: a specific child, a specific name, a specific and binding claim.
Commentary is from a charismatic Protestant perspective, drawing on KJV text and public-domain sources including Spurgeon, Andrew Murray, and Matthew Henry.