A family adopted him because his family were bad people, and they gave him their name; what are the consequences of that?

Firstly: 

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”

[al-Ahzaab 33:4-5]. 

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.