Is it permissible for a Muslim to give the Wi-Fi password he obtained from his neighbor to his family, or is its use restricted to him alone?

If you know that your neighbor allows your family members to use the internet, then there is no harm in it. However, if you know that he does not allow it, or you are unaware of his stance, then it is not permissible for you to do so except with his explicit consent. You can speak with him to ascertain his permission.