As another astronomer who studies compact stellar remnants like white dwarfs (I tend to focus on the even denser neutron stars and black holes), I thought I'd add my two cents.
If you look closely at the Science paper, you'll also find that they can't rule out that the companion to the pulsar is a Helium white dwarf, so it might not even be dominated by carbon/oxygen. We can also quibble about calling the low-mass white dwarf a planet, and in fact I had a long debate yesterday with a fellow astronomer about this very fact. While it's true that its mass is similar to those of planets, this is really the remnant of what was once a star. Its evolution to get to this point is extremely different from what most people (and astronomers) think of as planets.
Finally, I'll point out that careful examination of the original paper also shows that this may not be the only known such object. This pulsar (rapidly spinning neutron star pointed our way)+ white dwarf is one example of an ultracompact X-ray binary. (The most extreme example of these systems has a white dwarf orbiting a neutron star every 11 minutes!). Some of the other ultracompact X-ray binaries could also be carbon/oxygen white dwarfs with masses less than 13 times the mass of Jupiter (i.e., what some astronomers loosely define as the upper limit for a "planet").
I would not go so far as to call the press release idiotic. It has done a good job of bringing attention to this discovery, which is still extremely cool, even if "Planet Bling" is far from being true. Science communicators have a difficult job in walking the fine line between promotion and detailed argument.