I think the relevant piece is that he'd had that dream before.
It would be unusual if he had suddenly started dreaming about it, given we know he hasn't seen or heard about it in the last ten years. Although it isn't impossible, if something he saw or read reminded him of it.
It wouldn't be impossible that he dreamed of the flying motorcycle while he could still remember it - weeks or even days after that night ride. And as long as he keeps dreaming it, it can remain relatively "fresh" in his mind, remembering and dreaming of the dream he had a few months ago, more than the events ten years ago.
Another factor is, we don't know how much he might have seen or known about Sirius' motorcycle before being dropped off. If he had a lot of exposure to it, the motorcycle might have been a common dream feature, or helped fix flying on it in his childhood memories - that would make it more likely he would keep dreaming it long after he had forgotten the memories it drew from.
However, there are also indications he has actually been remembering that night all along in dreams - the nightmare about the green light and laughter he originally takes for the car crash, for example, which turns out to be a forgotten or suppressed memory, as seen in the extended flashback cause by dementors. That night as a whole is traumatic, I agree with @red_devil226 that it likely lingered in his subconscious because of trauma, much like stories of suppressed memories coming back in dreams for years.