Maybe it's the fact that since Serge is "supposed" to have died in the original timeline, when coupled with his role as the Arbiter, makes for a weird scenario involving him. Perhaps Schala was guiding him even.
Too many viable options for it, but there's very little to work with.
Here's my thoughts on the Frozen Flame thought by, well, Thought (heh):
Since prior to Serge's survival (i.e., the split point) both Another World and Home World have the same events, in both worlds Serge interacted with the Frozen Flame and became its Arbiter. Since the events which cause this to occur include Schala reaching out after hearing his cries, it may have been possible for her to interact through Lavos (and by extension, the Frozen Flame which is a splinter of it), and therefore interact with Serge. Extremely indirect I know. Perhaps she gave him a piece of her power or something like that, or an idea of what was going to happen.
If Kid is a clone of Schala, then if you were to follow the above paragraph to its conclusion, perhaps when Kid saved Serge, she inadvertently set off a chain reaction which left Serge with gaining things like this dream of the future. Something similar to the fact that Kid, being a clone of Schala and therefore somewhat similar (if not an exact clone, but the nature of how well the clone was made is debateable as Schala made Kid after she had already merged with Lavos, correct?), and also coming into close contact with Serge when she saved his life, might have triggered some sort of reaction to give him the dream.
Either that, or it's like watching a cold opening from a television show that makes you want to keep watching (or in this case, playing).
EDIT: Serge has the dream in Home World, but then the events happen in Another World, right? Perhaps that's a big factor into what is going on with this dream.
