Bruce gives him love, training, a purpose, and a home, but he also relies on him extensively for both physical and emotional support, and puts him in incalculable danger on a nightly basis.
Their father/son relationship is more co-dependent than one could say was always healthy. Bruce did in some ways look to Dick as a friend, though he's not the first parental figure to do so. But while Alfred might cook the food, Bruce is his legal guardian both in name and in the practical sense. They ate together, there's scenes where they more or less dressed together (in civvies) to go out together. Emotional support was understood to be one of the main things Bruce provided for Dick by understanding what he went through with his parents. There are times where Bruce reassured him emotionally. Alfred started running interference around the time the comics retconned in adolescent conflict between Dick and Bruce--which is why there are sometimes jokes about how Alfred is Dick's mother and Bruce is his father. I've no doubt Bruce is the one who taught him to shave and tie a necktie. Canonically Bruce signed his report cards and grounded him if his grades weren't up to snuff. The guy was very involved in all aspects of Dick's life.
Bruce is also responsible for a large part of Dick's ethical development via his training and one-on-one teaching. There is not a big divide between Dick's professional training and personal nurturing. They're Dick/Bruce, Batman/Robin and father/son.
Again, I don’t think the issue is adoption versus blood connection at all, since I do think that by the time Jason and Tim came along Bruce was ready to take on a more explicitly defined parental role, hence why he adopted Jason right away and Tim as soon as soon as he’d been orphaned.
But what's the difference between the relationships except for the piece of paper? Especially given that Dick was the kid he actually raised for most of his life? I doubt Tim or Jason would ever consider himself a son to Bruce where Dick was not.
But it’s something they’ve come to accept as it developed over time, not (to me) a reality that necessarily existed from the beginning.
Not from the beginning, no. I doubt Bruce knew what he was getting into in taking Dick in. Dick's the one who made him a father, though, through Bruce's raising of him. I don't think it took Bruce up until Dick was 23 to consider him his son. It just took him that long to admit it.