"Soon" is a very useful word for conveying ambiguity. If the speaker means "later today" or "several days later", they could use those phrases. Since they didn't, "soon" gives a sense of urgency or imminence without committing the speaker to any particular deadline.
To address your final question, "sometime soon", if anything, increases ambiguity. "Sometime", in this case, acts as an intensifier. Other potential intensifiers ("very", "really", "pretty", etc.) generically give the impression of seriousness, but "sometime" contains a reference to chronology. In fact, "sometime" can be used in place of "soon" to convey "it'll happen, but I won't commit to a timeline". Putting both together allows the speaker to communicate either immediacy or an indeterminate time in the future depending on how the listener wants to hear it.
You might think this sort of ambiguous way of speaking is dishonest. But it serves an important social purpose. It's not uncommon for people to have different priorities. In the sample dialog, A wants to talk and B (presumably) doesn't. A could ask, "But do you think I could come by later today?" That would put B in a place to need to say "no", which means both people are confronted with their different priorities. That might lead to hurt feelings or unease.
Instead A used the word "soon", which allows the other person to interpret the intention. Maybe they can find a compromise between "later today" and "never"? B might answer, "I have to finish something this week, but we could get together on Saturday". Not what A was hoping for, perhaps, but better than complete rejection.
To clarify, people don't usually put a lot of thought into this. "Soon" just rolls out of our mouths because we have an intuitive sense that it's the right word for the situation. Using other words or phrases isn't necessarily going to cause hard feelings. And in some cases, "soon" can be understood as dismissive because it doesn't commit to a particular time. As a rule, one word choice isn't critical since it's just one part of the context that people use to understand each other.