I think a big part of why they send you a skewer is that over time, the trainer can scratch up the pretty anodized finish on your regular skewer. Additionally, I think the included skewer is likely to fit the female skewer attachment(s) included with the trainer better than your regular skewer too (I know this was the case for me).

As far as using the skewer for normal riding... I wouldn't do it, but I have two reasons, and they are related to weight and aesthetics, not safety.

I think from a safety perspective, you would be just fine using your trainer skewer. The skewer that came with MY trainer, though, is considerably bigger (both heavier-duty and longer) than my Mavic skewer. I am quite sure it is much heavier; it's at least 25% longer. It's also silver when my regular skewer is black, and it doesn't match the front skewer (or the bike, which is also black)! These seem like trivial reasons, I know, but I wouldn't do it.