If you want to be his grandfather, and he wants to be your grandson, and you behave in a grandfatherly way towards him, and your daughter and his father have no objections, then you are grandfather and grandson.
The Queen's English supports this. The son of the Queen, Prince Charles, called Lord Mountbatten his "honorary grandfather". (Lord Mountbatten was his father's uncle.)
As for your daughter's siblings, they are the boy's aunts and uncles if they, the boy, your daughter and his father want them to be.
In English, the people involved choose the words in situations like this, and in the US, at least, it is nobody's business but their own what they choose.
(If your daughter and the boy's father divorce and you and the boy still have a close relationship, you can continue to be his grandfather if you and he want it.)
The legal ramifications (e.g., division of your estate if you die intestate) are beyond the scope of this site.