Yes, it is very common. It's called "using muscles you haven't used much before."
The muscles that run along either side of your spine are called the "erector spinae" group of muscles. They help your spine stay up. When you deadlift, row, squat, or anything like that, they get extra work - it's harder to keep your spine straight while you're holding an 80kg barbell than without it. After extra work, muscles sometimes get stiff and sore.
Just ensure you're using good form on your lifts. On all lifts, you want to keep your back in natural good posture. For the lower back, that means a slight curve inwards. Stand against a wall, and put one hand either side of your lower back. You should be able to have your bum and upper back against the wall, and to slide your hands freely in and out of that lower back space.
That's called the lumbar curve - lumbar's the lower back. You want to keep that while deadlifting, all the way from the first pull to standing upright, and all the way back down again. And keep it while rowing, and squatting, and so on.
At the top of the motion, consciously stick your arse out, and brace your abs as though someone's going to punch you in the stomach. Hold that brace all the way up and down with the lift. With the deadlift you'll need to pause and reset your brace with the weight on the floor, but should be able to hold the brace at the top of the lift.
Make sense?
If you don't keep it, then instead of your glutes (arse) and other muscles working, your lower back is working - and too much. You end up injuring yourself, and not developing the muscles you could develop.