Yes, they are one and the same. The name applied is usually determined by how the device is applied.
A double balanced modulator is just a double balanced mixer applied as a modulator. A classic example is the Motorola MC1496. It is listed as a balanced modulator/demodulator. But, if you look at the simplified schematic, you will see that it is actually a double balanced mixer. Page 9 of the datasheet talks about using it as a double balanced mixer.
Why the confusion? Why are they sold under different names? Who knows... Probably a marketing rather than an engineering decision.