No, using a "trainer" skewer on the road has no negative impact on your bike or safety. The skewer which comes with a trainer is mecahnically and functionally identical to the one that came on your bike, with the exception of the shape of the "heads" on the trainer skewer. It is actually identical to one of the original skewer designs. The reason that one comes with the trainer is because most bike manufacturers have changed the shape of the skewer "heads" that come on the bike to be more aesthetically pleasing. With the side effect that the skewer that comes on your bike is unlikely to fit securely into the clamp mount on your trainer.

No, using a "trainer" skewer on the road has no negative impact on your bike or safety. The skewer which comes with a trainer is mechanically and functionally identical to the one that came on your bike, with the exception of the shape of the "heads" on the trainer skewer. It is actually identical to one of the original skewer designs. 

The reason that one comes with the trainer is because most bike manufacturers have changed the shape of the skewer "heads" that come on the bike to be more aesthetically pleasing, which had the side effect that the skewer that comes on your bike is unlikely to fit securely into the clamp mount on your trainer.