Dear questioner,
Thank you for your important question.
May Allah, Most High, reward you for your desire to increase in knowledge.
The child that is born out of wedlock is ascribed to the mother and not the father. Hence, the boy is not your brother.
If your father was married to the mother of this child, then he is your half-brother.
The relationship of unmarriageable kin (Mahramiyya) is established in three ways; kinship, foster relationship, and relationship through marriage.
If this child was not breastfed by the same wet nurse you had, he is not your brother through a foster relationship.
If this child is not related to you by marriage, he is not your brother.
Yes, this child was born from the same father. However, due to your father not being married to this child’s mother, the child is legally not his son and cannot be attributed to him.
As long as the child has not reached puberty, you may interact with him as you do with all children. Once the child reaches puberty, You must observe all the rules and decorum of dealing with the opposite gender when it comes to this child.
(Kasani, Bada`i al-Sana`i)
This child is your ‘brother in Islam’. You should grant him the same rights you would grant your neighbor or friend, such as helping them when they are ill, replying to their greeting, etc.
Please see: Who is a Mahram.
May Allah, Most High, raise your status in paradise for your worry and care of others.
And Allah alone knows best,
[Ustadh] Omar Popal
Checked and Approved by Shaykh Faraz Rabbani