You're correct in assuming that pole dancing is an excellent workout for the upper body. However, pole dancing has the potential to be an excellent total body workout as well. The main thing to remember is that the specific muscles that are being worked, and how hard, really depends on what kind of moves you're doing on your dance pole.
When spinning and climbing, the muscles of the upper body are obviously being used to hold your body weight on the pole. Specifically, the lats (large muscles along the sides of your back), shoulders, biceps and forearm muscles are called into play with these types of moves.
The core muscles also play a very important role in pole dancing, particularly during inverted (upside down) moves. This is especially true when the exit or transition from the move involves reversing the inversion in what is essentially a reverse crunch. This is because the forces of gravity act as resistance during such moves.
Lower body strength is developed with moves that require primarily leg strength to hold yourself on the pole, such as hands-free poses on the pole. Climbing also requires some leg strength, in much the same way that rock climbers use their legs to assist in lifts.
Beginner level pole dancers who aren't yet ready for these more advanced moves can also get an excellent strength workout however. Many beginner level spins require considerable upper body and core strength, especially for someone who is new to this form of dance.
Even simple moves that don't require both feet to leave the ground can work the core and lower body muscles. This is because pole dancing is typically done in a relatively slow and controlled manner, which tends to help the dancer focus on muscle control and engagement.
There are also numerous exercises you can do on a dance pole to build strength in preparation for higher level moves. In other words, you can also use your dance pole as a workout tool to prepare you for more advanced moves, much like you would use a piece of gym equipment.
Something important to remember about pole dancing is that even though it's a fun form of fitness, it's still a workout. Like all other dancers, pole dancers should make sure to warm up sufficiently before dancing, and particularly before attempting more advanced moves.
Overuse injuries, particularly in the shoulders, are fairly common in newer pole dancers. This is likely due to "too much, too soon," so it's important to pace yourself and to stop before you feel tired or sore. As with any workout routine, you will build strength over time and it's best to take a reasonable and gradual approach to avoid injury.
Overall, pole dancing is an excellent strength building activity, and many women find it much more fun and engaging that lifting weights at the gym. When done with care and attention, along with a view to gradually increasing difficulty, pole dancing can be a fabulous total body workout.