The more I think about it, "reptilian brain" is probably a very ambiguous phrase. There are definitely similarities between the whole brain of a reptile and the inner structures of our brain, and the evo/devo story (the links between evolution and development) contributes to that view.
But there's two problems with the terminology I think. One is that we didn't evolve from the reptiles of today. Both us and reptiles of today have a common ancestor. The terminology may lead to the former view.
And the other is that it lacks a rigorous definition. It most likely applies to the homology between reptiles and mammals (as defined by our last shared ancestor mentioned above) of some particular set of structures. The basal ganglia was referred to as the "reptilian complex" by MacLean; some aspects of his theory were eventually shown to be wrong. That is, reptiles are now thought to have always had a neocortex-like structure (the dorsal cortex) ... it's just really tiny compared to human cortex.