For starters, there is no definitive answer to anything in Ulysses, but I do really like your question. It provokes a number of things for me that point to one of the main themes in the book: Parent and Child, and in this case in particular - Father and Son.
Bloom often reflects, throughout the book, on his (and Molly's) own lost son who died young. And also in a metaphorical way, Bloom reflects on his own inabilities to be a protector, a father, and even in many significant ways, a "man" (according to the standards he encounters throughout his day).
Stephen's father is a drunk who wastes the family's money. So, in a base way, a very main part of the book (and indeed also in its references to the father son dynamics of Jesus/God, Odysseus/Telemachus, and Hamlet/his father) is the search for the lack that a son-less father has and a father-less son has.
In the book, especially in this chapter, we see the possibility of Stephen getting a father and of Bloom getting a son (and in turn maybe making his relationship with Molly stronger). I think in the end perhaps we can say that Stephen realizes at some point that Bloom is NOT his father and can never be his father. Perhaps his sobering up has made him realize this. Perhaps Stephen realizes as well, like the narrator of Joyce's "Araby," his own immature yearning to fill that lack of a father figure with a stranger. This is one answer that you could explore, and I think it's really key to the entire novel as well. Thank you for the question, I enjoyed thinking about it and writing this!
