Is there a (realistic) way for such a planet to exist?
The minimum exposure time for such a photo is about 10 minutes (360 degrees in 24 hours means that to have 15-degree arcs you need one hour of exposure; five minutes will yield blurred arcs of less than three degrees).
The scotopic retinal persistence (bright lights seen at night) has a relaxation time of about 1/10", slower than the normal which is around 1/25" but still too much.
You would need to compress those 10 minutes (600 seconds) in one tenth of a second, which means a rotation six thousand times faster than Earth's - one revolution every 15 seconds approximately, or 4 rpm.
At that speed, even if the planet is much smaller and denser than Earth, on the equator (let's say 3000 km radius) you get a centrifugal acceleration in excess of fifty thousand G (centripetal acceleration is proportional to the square of the rotation frequency. Six thousand times the speed gives you thirty-six million times the force; even dividing the radius by two leaves you with an eighteen million times increase). The planet would disintegrate almost instantly.
You cannot simply increase persistence of vision by positing a really long scotopic decay alone, because we're dealing with a factor of six thousand; at that point, just moving your head would completely compromise your vision. You would need to move thousands of time slower. But this means that days will flash by, and you'd see the sky blazing on and off. So, you need also a perpetual night (at least in the inhabitable zone of the planet).
You could get something like that, therefore, on a very small planet where humans lived in slowed down time (e.g. Charles Sheffield's Sight of Proteus, and Between the Strokes of Night again by Charles Sheffield). If their "day" actually lasted sixteen years, they would see a 24-hour day's night sky like that. Of course you would need to have no day at all, so either make it very small and tidally locked to a very near, very dim star (or a black hole - e.g. in Greg Egan's Incandescence), or have it be a rogue planetoid far away from any star.
The scenario would be halfway between Between the Strokes of Night and, perhaps, the backstory of Blish's Surface Tension: humans found themselves on a very low gravity, perpetually dark planet. There are some biological reactions available but energy gathering times are too long to sustain human life (the energy in a field of wheat requires one year's solar output and can feed X humans for one year. Lower the energy due to darkness, and X drops so low that you no longer have a population).
So their solution is to cheat. By slowing perceived time using Sheffield's S-Space by a factor of six thousand, the bioengineered humans will receive six thousand times more light, and be able to see. And the sluggish semiorganic crops of the land will grow six thousand times faster, supplying the raw materials to produce everything else for a population that requires six thousand times less food and oxygen. Also, the negligible gravity of this asteroid is apparently multiplied (9.81 meters per second squared, but a second is now almost two hours) and - handwaves quickly - is now almost exactly Earth normal.
Edit: fast planet.
What if, instead of the planet rotating faster on its own axis, the speed at which the planet orbited its star would be increased by a massive amount? Could mass of said star, the planet's distance to it and the speed at which it orbits the star be set in a way that allows for stars appearing as lines?
The rotation has still more or less the same speed - we need five degrees' worth of rotation to take place in less time than it takes the perceptual image to fade. The planet has to rotate very fast around its primary. This is along the lines of pulsar planets (one example is PSR_J1719-1438 b, aka the "Diamond Planet" - it's thought to be the compressed diamond core of a shattered star - now how cool is that?). In truth, any planet orbiting near enough a sufficiently massive object will do.
The problem is that the planet needs to be even nearer and/or the star heavier than the pulsar case, which places the planet inside the accretion disk of a probable neutron star or black hole; indeed, this latter one is the exact scenario of Greg Egan's Incandescence. It is very doubtful that the overall conditions, primarily the radiation flux density, would allow people to stay in the open and look at the stars.
But if they could and did, yes, they might see something like the image you seek. I strongly suspect that whatever they saw, it would manage to be even more mind-blowing. Parallel to the "axis of the heavens", they might see the Blue Ghost of Larry Niven's Smoke Ring. And all around, the Highway of the Gods.