The underlying oddity here is that the ship cannot be going fast enough to actually pass any stars like that - that would essentially mean going hundreds of thousands times faster than light, rather than just hundreds or thousands times as the plots imply. And never mind that the streaks appear even when ships do short warp hops within a single star system!
On the other hand, the streaking things do appear to be stars - at least some of them coalesce into stars when the ship drops from warp to impulse, and elongate from stars to streaks in the opposite case.
Perhaps the warp field is pulsating around the ship and distorting our view of the surrounding space, so that the same stars go "past" the ship again and again and again? And perhaps the pulsation is a bit faster during acceleration than during cruise, much like a locomotive engine might have its wheels slip and turn like mad for a few moments before they grip the rail and start moving the train relatively slowly.
I thought about this the other day, and how if a ship really was going that fast, it would look like it was at impulse power, because there is no way they are passing that fast. Then I thought about something:
Could it be possible that an observer would see some kind of streak because he's travelling faster than c, while the light from the star would be going at c only?
Or may be, the warp effect that squezes the space around the ship would cause the stars to elongate. Sort of like how the ship is long when it first jumps? I don't know...