I'm reading through the whole interview (that was a really good dialectical move, here, by the way: putting the alert at the head of the OP made me feel prospectively guilty about the thought of responding to the OP without reading the interview), so I'll respond in terms of this reading. Here is one earlier, highly relevant section:
Mathematics is a practice whereby you fully transcend yourself. You get the skills and the demonstrations, and at first you don’t understand them. Then, later, light begins to slowly illuminate the completely objective topics. Mathematics is never about the mathematician; philosophy is always also about the philosopher. It involves the thinker of those thoughts, whereas mathematics does not presuppose the thinker. Another difference is that in philosophical training, it’s important that the student understand the nature of every stage at that stage. In mathematics, conversely, it’s very important that you don’t understand the nature of the stage: just follow the rule blindly. That’s part of the skill.
Although probably (or almost definitely, as in the case of someone like Cantor) overstated, I think that he's getting at something true enough, and hence admitting of insightfulness enough, to contribute to his discourse on thought-as-a-sense and fields-of-sense further along. All this talk of stylish mathematics reminds me directly of the matter of mathematical style as a philosophical topic and then of the relationship between syntax, semantics, and semiotics in the domain of generalized metalogical discourse.
He soon says:
My books are, as it were, accessible math books. Right now, I’m working on an accessible “philosophy of quantum mechanics” book, to figure out to what extent the deep mathematics of quantum mechanics can be explained in this way. Maybe they can’t. I don’t know yet.
That's a good sentiment to have, here ("good" as in virtue-epistemological terms, say). Also, in light of it, I suspect that the fields-of-sense concept is adapted from quantum field theory in some way. I'll see if that's so, but it was what I was struck with right from reading the phrase "fields-of-sense."
Next, the "crux":
We tend to draw distinctions between sensing and thinking, and the reason for this goes as far back as Plato and Aristotle. Here’s how the idea usually goes. Sense organs have their proper object: smelling is for smells, hearing for sounds, seeing for colors and shapes, and so forth. They have their proper objects, whereas thinking unifies these objects. If I think of a particular wine that looks yellow and tastes a little bit citrusy, then my thinking unifies the citrus taste and the yellow color. The idea was that since thinking synthesizes different sense impressions, it cannot be a sense organ itself — but that’s a straightforward non sequitur! It assumes that a sense organ is a sense organ because it has a proper object that can be synthesized in thinking. Why not assume instead, and this is my starting point, that thinking is an additional sense organ? Namely, the synthesizing sense organ, which is something that Aristotle explicitly entertains when he coins the concept of common sense (koinē aisthēsis in Greek, which will then become sensus communis in Latin, and “common sense” in English). The very idea of common sense, for 2,400 years, has been exactly the notion of thinking as a sense. That’s Aristotle preserved in the English language. I accept this idea, but I say that Aristotle made a mistake when he concluded that thought is not a sense. He should have developed and argued for the idea that thought is a sense — something he entertains in only two places in his work. So, what does this mean? How do we then understand a sense? A sense is a fallible way of being in touch with a mind-independent reality. In thinking about mathematics, we are in fallible touch with a mind-independent reality. If I make a calculation mistake, which is very easy in complex mathematics, then I get reality wrong. If I miscalculate a linear operator in quantum mechanics, I just make a mathematical mistake. Why not think, then, that I’m in touch, as a thinker, with a reality that cannot be grasped by any other sense?
On the one hand, again, it's a strong move on his part. You can frame what he's saying as a rejoinder to skepticism about senses like seeing. In other words, if even thinking is a sense, and the qualia of its sensations are things like syntactic relations or other metaphysical jargonified terms, then by some kind of anti-skeptical parity about the value of thought, we can see the value of seeing a priori anyway. (In fact, there was a whole debate around the time of Frege about whether the laws of logic, as laws of thought, represented necessary psychological truths, but still psychological truths; and J. S. Mill, for example, traced the law of noncontradiction to the sensation of truth-nullification involved in asserting and rejecting the same belief.) However, from his tone directly, I think his more crucial point, and error, is about how "the presupposition" of the non-sensory character of thought is a presupposition.
In Kant, for example, it is not that the senses are demarcated from thought on an intuitive basis, but he explains the dynamical reason for their differentiation on the relevant conceptual level. (For in Gabriel's terms, we can still differentiate thought from other senses, and even dynamically, perhaps; but he is using the word "sense" in a different sense(!) than Kant, then.) Sensations are passive or reaction, whereas thoughts are, as the procession of the understanding and reason, "spontaneous." You can critique the arguments and considered judgments that Kant builds this talk of spontaneity on, but I don't think it'd be fair to say that Kant was just presupposing that thinking is not a sense. Trivially or not, by his definitions, the word "sense" is better reserved for the deliverers of information like sight and hearing, and the interpreter of information, albeit in the spacetime forms a quasi-source of information in its own right, is then not the same as to be such a deliverer. If it were, Kant said, then such an entity would be equipped with intellectual intuition, which is the form of divine creation in abstracto. This is the ultimate reason why, in Kantian terms, thought is not a kind of sensation.
However, note that the whole strongly modern question of defending philosophical analysis by appeal to "intuitions," whether those of "trained philosophers" or the philosophy laity (the human laity overall, in fact), testifies to the residue of a conflation of the dynamical form of thinking with the dynamical form of sensing.
Now, Frege had the interesting idea that, given the plurality of isomorphic geometries, it would be possible for everyone in the universe to perceive reality according to a different geometrical theory, whereas their logic on their perceptions would not be so variant. This is in line with what Gabriel emphasizes about talk of common sense, as thought, here. Thought-as-an-objective-sense, or a more purely, fundamentally intersubjective one.
A further specifically interesting/relevant point in the interview:
The lowest hanging fruit is the recognition of what I call self-evident moral facts. They’re so obvious that we don’t usually dispute them...
Here, he sounds like a moral intuitionist, which is again an example, in the history of philosophy, of a thought-sensation merger perspective. Many, or maybe even all, moral intuitionists (not so called) before G. E. Moore were empirically minded about how this faculty operated, or what it operated upon, but Moore allowed that we could intuit certain facts of goodness through abstract speculations about worlds empty or full of certain things in certain ways (a world empty of consciousness but full of beauty, for the example I feel I can remember from him).
And as to fields-of-sense:
This is a crucial point. We need to reduce the transfinite, or what I now call “the hypercomplexity of fields of sense.” Reality is not just a form of complexity: it’s hypercomplex, and no mathematical model will be able to represent reality as such.
I really do think he has the relevance of QFT as a picture of objective reality in mind. QFT can be seen as a variation on the theme of the geometrical grounding, now the topological grounding, of macro-level objects: not point particles or even quite so much detached strings, but the algebraic zone whose localized solutions are functional equivalents of point particles, strings, branes more generally. He mentions an "algebra of normativity" in connection with this thematism, obliquely or not. But anyway, then a sense can be modeled in terms of fields, mathematically no less, but modulo hypercomplexifiers; and on this analysis of what the word "sense" means, it does make a lot of sense to interpret thinking as a sense. Sight is a field of color and geometrical information; hearing is a field of sound and geometrical/topological information; perhaps thought is a field of syntactic qualia and just topological, not "also" geometrical, information.