Our Gemara on Amud Beis discusses the legal entitlements to certain properties and materials that unwittingly became shared by two potential beneficiaries and investors:

שָׁטַף נָהָר זֵיתָיו וּנְתָנָם לְתוֹךְ שְׂדֵה חֲבֵירוֹ, זֶה אוֹמֵר: ״זֵיתַי גִּדֵּלוּ״, וְזֶה אוֹמֵר: ״אַרְצִי גִּדֵּלָה״ – יַחְלוֹקוּ.

In the event that a river swept away one’s olive trees and deposited them in the field of another, and they took root there and yielded olives, this one, i.e., the owner of the trees, says: My olive trees yielded the olives and so I have a right to them, and that one, i.e., the owner of the field, says: The nourishment from my land yielded the olives and so I have a right to them, then they divide the olives between them.

Penei Dovid (Shemos 3:2) uses this Halacha to metaphorically understand the institution of adoption. 

The Torah does not have a formal definition or status for adoption of a child. For comparison, the social institution of marriage has precise legal definitions and rules for how it is established and dissolved according Halacha. However, the Torah has no formal system for adoption, nor for termination of parental rights. Yet, the social institution exists because, out of sheer practicality, there will always be situations when parents are deceased or unavailable and some neighbor or relative must fill the role. The Gemara (Megillah 13a) tells us:

 כׇּל הַמְגַדֵּל יָתוֹם וִיתוֹמָה בְּתוֹךְ בֵּיתוֹ — מַעֲלֶה עָלָיו הַכָּתוּב כְּאִילּוּ יְלָדוֹ.

Anyone who raises an orphan boy or girl in his house, the Torah gives him credit as if he gave birth to him.

One who adopts ought to receive merit in accordance with our Gemara, where the owner of the host field and the owner of the olives have equal share in the resulting produce. This reward might have been shared with the biological parents, as they supplied the genetic material. Nonetheless, Hashem grants as a bonus, the full reward as if the adoptive parent was also the biological parent. (I suppose the biological parents still get their reward due for their role.)

The idea of adoption is also on a continuum. For example, the Gemara (Sanhedrin 19b) states: “Anyone who teaches another person’s son Torah is given credit as if he fathered him. But even in smaller ways, if one is a mentor or role model, this too is a kind of adoption. As social beings, we are interdependent and interrelated. We share in each other’s fate and have many opportunities to be a part of another person’s life. It is not just something to do, it is also something that becomes an extension of our own agency and destiny. This is similar to the idea stated in Bereishis Rabbah (30:6 and Rashi Bereishis 6:9): The main progeny of righteous people are their good deeds. What we do in service of others creates generational impact, just as having children does.