It's crazy how many different things are mashed together in this "story".
An anomaly that breaks a conservation law is a
very familiar concept in quantum field theory. Sometimes it allows us to rule out a theory, other times it produces an observable effect. The most famous example is the
chiral anomaly, which contributes to the decay of the pion into two photons.
Pions and photons are a topic for particle physics. Meanwhile, quantum mechanics applied to the collective behavior of electrons in solid matter can produce so-called "quasiparticles", and some of these can also experience the chiral anomaly. This has been studied for a few years. The theoretical understanding of the observed phenomena seems to be incomplete, to say the least.
The chiral anomaly for quasiparticles is a bit different to the chiral anomaly for elementary particles, in that you could say it's due to an approximation - what is called an "effective field theory". As is explained in figure 1
here, the chiral anomaly for quasiparticles appears just because the approximation doesn't include certain transitions that nonetheless occur. In a slightly more complete description of the physics, there is no anomalous nonconservation.
In any case, the mathematics of "chiral gauge anomalies" that was first developed in particle physics, also applies to the quasiparticle anomalies of condensed-matter physics. But in particle physics, there is a slightly richer range of chiral anomalies that can occur.
Described in terms of the triangular Feynman diagram that is the ubiquitous symbol of the chiral anomaly - in which you have a chiral fermion tracing out a triangle, and a number of bosons coming off the triangle - in the gauge anomaly, all the bosons are gauge bosons; in the gravitationa anomaly, they are gravitons; and in the mixed anomaly, you have both types of boson contributing.
The gravitational and mixed anomalies are not directly relevant to quasiparticles. They should only be observably relevant in extreme conditions of particle physics, like the early universe or maybe a neutron star.
However, the mixed anomaly in particular, can be worked out in a way which emphasizes just the gauge aspect or just the gravitational aspect (or a way which retains both). I am not certain, but I suspect that this double nature of the mixed anomaly is the mathematical avenue which has allowed it to be used to describe condensed-matter chiral anomalies in which only gauge forces are physically present.
Whatever the details, it is certainly true that condensed-matter physics is full of phenomena which are being dubbed analogues of phenomena from elsewhere in physics - for example, one can read about "black hole analogues" - and sometimes the physicists don't make it clear that they are not talking about the original thing. For example, a few years back there was a story about observation of magnetic monopoles, which would have been a breakthrough in particle physics, but in fact they were just analogous objects.
So the thing which has irked me the most about this story, is that it is another example of analogous phenomena being described as the real thing. There is no actual mixed gravitational anomaly being observed here, it is at best a quasiparticle gauge anomaly that is mathematically analogous to a mixed anomaly.
I say that without having plumbed the depths of the theory involved - it's just common sense - if someone wants to argue otherwise, please speak up. Anyway, some of the reporting about this experiment is pretty clear that we are talking about a tabletop analogue of the true gravitational anomaly. But
the scientific paper itself is thoroughly opaque on this issue. It keeps saying that (e.g.) the mixed anomaly has been "tied to" these more down-to-earth, purely electromagnetic processes, "even in flat space". A casual reader has no chance of realizing that this is just analogy.
Furthermore, there isn't even agreement that this particular form of the chiral anomaly (the analogue mixed anomaly) is actually at work here. Some of the other experts in the field simply disagree with this interpretation of the observations.
As for the role of string theory - like the chiral anomaly, string theory is fundamentally meant to explain elementary particle physics, but its mathematical techniques can be employed to describe quasiparticles too, and apparently it played a role in developing the theoretical analogy between the mixed anomaly and its quasiparticle analogue.
I don't normally recommend Peter Woit's anti-string blog, but I first sorted out all this in
the discussion there, where you can also see comments from some of the theorists involved.