Praise be to Allah.
At the beginning of Islam, adoption was permissible and the adopted child would be named after the one who adopted him. Then Allah, may He be exalted, forbade it and ordained that everyone should be named after his (biological) father.
Allah, may He be exalted, said (interpretation of the meaning):
“nor has He made your adopted sons your real sons. That is but your saying with your mouths. But Allah says the truth, and He guides to the (Right) Way.
Call them (adopted sons) by (the names of) their fathers, that is more just with Allah. But if you know not their fathers (names, call them) your brothers in faith and Mawaleekum (your freed slaves). And there is no sin on you if you make a mistake therein, except in regard to what your hearts deliberately intend. And Allah is Ever Oft-Forgiving, Most Merciful”
It is not known that there was ever any difference of opinion among the scholars concerning this ruling.
It says in al-Mawsoo‘ah al-Fiqhiyyah (10/121, 122):
Islam forbids adoption and deems all its consequences to be invalid. That is because Allah, may He be exalted, says (interpretation of the meaning):
“nor has He made your adopted sons your real sons” [al-Ahzaab 33:4] and “Call them (adopted sons) by (the names of) their fathers” [al-Ahzaab 33:5].
Adoption was known among the Arabs during the Jaahiliyyah and after Islam came. During the Jaahiliyyah, if a man admired another’s man strength and looks, he would take him as a son and give him his name, and give him the share of one of his own sons in terms of inheritance, and he would be attributed to him as “So and so [the adopted son] the son of So and so [the adoptive father].” The Messenger (blessings and peace of Allah be upon him) adopted Zayd ibn Haarithah before Allah honoured him with the message, and he used to be called “Zayd ibn Muhammad”. This is how it remained until the words of Allah, may He be exalted, were revealed: “nor has He made your adopted sons your real sons... And Allah is Ever Oft-Forgiving, Most Merciful” [al-Ahzaab 33:4-5]. Thus Allah abolished the system of adoption and instructed those who had adopted someone not to attribute him to himself; rather he was to be attributed to his (biological) father, if he had a father who was known. If his father was unknown, he was to be called “mawla” or a “brother in faith”. Thus the people were forbidden to change realities, and the rights of inheritance were prevented from being lost or reduced. End quote.
Based on that, that family must hasten to set straight that which they had done wrong. They have to cancel your attribution to them in official papers and documents, and they must set things straight in accordance with sharee‘ah. That is to be done by testifying to your real lineage and confirming that with documentation to that effect. The fact that the parents are bad people is not an excuse to deny one’s blood relationship to them.
From this you will know the answer to the basic issue of the question, which is that if a family adopts a male or a female, there is no blood relationship between them as a result of that adoption. Based on that, they are non-mahrams and it is not permissible to interact with them as families interact, according to the rulings. The woman who adopted you is a non-mahram to you, and it is not permissible for you to see her, let alone touch her or kiss her when you are not a mahram to her or to her daughters.
The Prophet (blessings and peace of Allah be upon him) married the ex-wife of his adopted son Zayd – namely Zaynab bint Jahsh (may Allah be pleased with her) after Zayd divorced her. This was a practical application on the part of the Messenger (blessings and peace of Allah be upon him) of the ruling that annulled the effects of adoption.
Yes, if that woman who adopted a child breastfed the adopted child five times before the age of two years, then he would become a son to her through breastfeeding, and her husband would become a father to him through breastfeeding, and their children would all become brothers and sisters to him through breastfeeding. But as that did not happen – according to what you say – then that woman is a non-mahram to you, and she must observe hijab before you and interact with you as a non-mahram woman would.
You should honour that family and take an interest in their affairs and keep in touch with them, because of the favour they have done to you.
For more information, please see the answers to questions no. 4696, 6102 and 95216.
And Allah knows best.