When the Narnians talk about being slaves, I feel like they're talking about being trapped in the world of the Calormens. Narnia and Narnians are supposed to be a metaphor for the Christian world and its people. In the Christian faith, you are not supposed to be a "slave" to the superficial, flesh driven and unholy ways of the world. The Calormens and Tashbaan represent this unholy while Narnia represents the eternal freedom that comes with Christianity. So, for Bree being a Calormen or living in that world makes you a slave while being a Narnian makes you free. Also, for Bree and Hwin to be taken from Narnia and forced to live with the Calormens is to rob them of their freedom and force them into a world of slavery. It is the same for Susan and Aravis. They are both threatened by the possibility of being stuck with the Calormens forever through the bond of marriage. The Calormens evil ways are heightened by the fact that they also have actual slaves. This is pointed out several times I think to remind us that to be a Calormen or choose to stay in their world is to belong to a world of slavery. From what I gather and remember, Narnia does not have slaves because the people there are free.
I think this is an interesting point! With this book being the third in a series where the first two are incredibly thinly-veiled allegories for two of the most important Christian stories: Genesis and the death and resurrection of Jesus, the Christian aspects must come into play somewhere. The book itself is about a journey–or quest, should I say–to escape the lives that our main characters didn't want, which in and of itself could be seen as a form of slavery. I feel like this parallels nicely with the "slavery" that all humans must endure being creatures prone to sin, and they must go on a spiritual journey to escape that life.
Agreed with Catherine here--this is a really sharp point and useful framing. In that sense, I think Bree's attachment to his status as "talking horse" and therefore more free becomes even more interesting because we see the way that that particular investment actually becomes a form of worldly attachment.