On the other hand, if you do take your fursona very seriously, perhaps you wake up at odd times because you think you should. It seems weird, but you can actually control sleep more than you might think. For instance, I used to have an alarm clock that I used whenever I needed to be up early, and I got to the point that if I set it, I would wake up before it went off each day, by a few minutes. Sometimes I even found the power had gone out in the night and the clock was reset, but because I expected it to go off, I woke up anyway.
I also tend to sleep quite heavily, and move around a lot in my sleep. Yet, when I'm living on the streets, very quickly I'll start waking up the moment I sense anything moving near me, and I've slept in places like trees without falling out, and on the tops of concrete pipes raised off the ground in order not to get wet, and manage to stay on it all night even though it's round and I usually roll all over the place.
This is all anecdotal of course, but I don't think some people realize just how aware of things we can be in our sleep. If this is something like... you found out your fursona should be awake at night, and suddenly you start waking up in the night, it could well be a mental thing.