That's correct. In the screenshot above, please note the text in the green box. It doesn't say that you will get a Rare bear. It also doesn't say that it will give Rare bears +6 levels; it says that it will give Rare bears +6 max levels. What does that mean?
All non-Common bears have a limit to the number of levels they can gain via Bear School. E.g., your Rare bears might each be allowed to gain only 12 levels via School before being prevented from any additional schooling. The way to increase this limit (e.g., to let your Rare bear gain 18 levels rather than 12 before being prevented from additional schooling) is to do School with the bears of the rarity just below (e.g., the limit on Rares is increased by schooling Commons, the limit on Epics is increased by schooling Rares, etc.).
That's what is described in the screenshot shown above. After you've done Common Bear School 25 times, the limit for Rare bears goes up by 6.
In summary, you don't gain a new bear, and you don't gain more levels for your bears. You do increase your bears' potential to grow by increasing their max levels.