Is it permissible for him to adopt a girl whom his wife had adopted when she was alive?

We have previously explained that the word adoption, as customarily used by
people, may mean two things: 

1.    
Undertaking to raise a
child and care for him, whilst attributing that adopted child to the
adoptive family and regarding him as one of them.

2.    
Undertaking to raise a
child and care for him, without changing his name (i.e., still attributing
him to his original lineage). 

The former was permissible at the beginning of Islam, then it was declared
invalid by the Qur’an. The latter is permissible and it is prescribed in
Islam.

See the answer to question no. 126003 

If the adoption of that girl was of the first type, whereby this girl was
regarded as being the same as the couple’s other children in terms of
lineage, inheritance and mahram relationship, then it is haraam and it is
not permissible to continue it, whether the wife has died or not. If your
wife did that during her lifetime, it is not permissible to continue that
after she dies. 

But if it was of the second type, there is nothing wrong with it; rather it
comes under the heading of favours and kindness to other people. If the
adopted girl was also an orphan, the Prophet (blessings and peace of Allah
be upon him) said: “I and the one who sponsors an orphan will be like this
in Paradise,” and he gestured with his index finger and middle finger.
Narrated by al-Bukhaari (6005) 

So carry on with what you are doing of taking care of this girl and
sponsoring her, without attributing her to yourself or making her an heir
like one of your own daughters, or regarding her as one of your mahrams who
can uncover in front of you, or with whom you can be alone, or who can
travel with you like one of your own daughters, once she reaches puberty or
is close to it, and becomes a woman who is regarded as desirable. All of
that is not permissible.

There is nothing wrong with you marrying her, if that is appropriate, or
with one of your sons marrying her, if you have a son. 

Please see the answers to questions no. 5201 and
185184 

And Allah knows best.