I think you have to take that leap of faith and increasingly treat them as your own. My son, now divorced, took on four step-children from 4 to 12, when he married ex-DIL. He treated them as his own, and one of them changed her name to the same surname when she was 20. All of them have stayed in contact with him, and they are all adults now. When the marriage ended, they were all confused and angry with their mother, who demanded they take sides, starting with not maintaining contact wth me and reminding me that the children I had treated the same as my grandson were no relation. I do see them every now and again, but it was heart-breaking to experience the damage that was done, especially as the one who changed her name now has 3 chidren herself, and her young ones saw me as a permanent fixture in their lives. It was during that year that I realised how much we had been through together over 12 years, and I could not help but grow to love them. They all know I still care about them, but I have been made to understand by ex-DIL that I can see my grandson but should not nvolve myself with them. They don't know their own grandparents, who were ousted immediately her previous two relationships ended.
Children do not weigh up whether you are legally related. They all need to feel safe and loved in equal measures, whilst they are growing up. They can work out family dynamics when they're older, and boast how well they were treated.