The term peculiar references that they were alone in their sanctioning slavery...which they were not...so you consider Deleware and Maryland to be a southern states.
They were nominally southern or southern leaning in portions of both states.
The use of “our own” is the same “...they were alone in their sanctioning slavery...” The meaning of the term peculiar in the full context of their usage by both Calhoun and Stephens in speeches indicates that they saw it as an institution that was southern in nature.