About two years ago, I spent the better part of a month trying to get to the bottom of this question: What is a Frobenioid? After all, the IUTer's are fond of insisting that it's incumbent upon other mathematicians to invest time in understanding the foundations of IUT simply because Mochizuki's claimed theorems are, if true, so remarkable. I'm not a number theorist, so I had little hope of understanding the main body of IUT, but the foundational material on Frobenioids is pure category theory, so why not give that part a go?
I came away feeling that my time spent on this project was wasted, for several reasons:
It was around the same time that Scholze and Stix met with Mochizuki. Apparently they were able to agree from the outset of their discussions that the whole apparatus of Frobenioids was unneccessary for understanding IUT. In this light, the insistence of the IUT camp that one can only understand the theory by plowing through the foundational papers from the bottom up (dubious as it may ever have seemed) is downright insulting. Essentially, I was promised mathematical insight for reading Mochizuki's papers and then told that I was a sucker for taking this promise at face value.
The definitions found in Mochizuki's Geometry of Frobenioids I, II are byzantine. My opinion by the end of the project was that the whole apparatus, with a few minor tweaks, could be much more cleanly expressed, with much less need for ad hoc terminology, by using basic properties of the language of Grothendieck fibrations. Once re-expressed this way, the definitions appear clearly to consist of a somewhat-reasonable core, encrusted with numerous ad hoc extra conditions. I find it bewildering that Mochizuki did not care to use this clarifying language when writing these papers -- are not Grothendieck fibrations standard fare for anybody who is familiar with the material of SGA?
It might be argued that Mochizuki's definitions as stated are "more correct" than the tweaked versions I ended up formulating for theoretical reasons to do with how they are used in IUT. But my distinct impression is that this is not the case. To the extent that Frobenioids are used at all (and as discussed above, their use at all is apparently inessential), it is only certain well-behaved Frobenioids which are ever invoked, and these fall squarely within the parameters of the tweaked definitions I ended up using.
The avoidance of fibrational language is not only obuscatory, but downright harmful. In the course of this project, I found a counterexample to Proposition 4.4.ii in The Geometry of Frobenioids I, while also finding an amended statement of the theorem which is in fact true. I wrote to Mochizuki about it, and he kindly took my objection seriously. However, his response was to add a note to one of his many online manuscripts detailing errors in his work which corrected the statement, without engaging in a serious way with the material. In fact, I assert that the claimed theorem, in amended form, takes more work to prove than appears in the paper or the errata.
Anyway, here's what a Frobenioid is, in slightly tweaked form:
Definition:
A commutative monoid $\Phi$ is said to be divisorial if $\Phi$ is cancellative (i.e. $x+y = x+z \Rightarrow y = z$), group-free (i.e. $M$ contains no nonzero invertible elements), and saturated (i.e. $M$ is closed under division in its group completion $\Phi^{gp}$).
If $\Phi$ is a commutative monoid, let $\Phi \rtimes \mathbb N_{\geq 1}$ denote the semidirect product of $\Phi$ with the multiplicative monoid of positive integers, with respect to its canonical action.
If $\mathcal D$ is a category, then a divisorial monoid in $\mathcal D$ is defined to be a functor $\Phi: \mathcal D^{op} \to \mathsf{CMon}$, where $\mathsf{CMon}$ is the category of commutative monoids) taking values in divisorial monoids. [1]
If $\Phi: \mathcal{D}^{op} \to \mathsf{CMon}$ is a divisorial monoid in $\mathcal D$, then the elementary Frobenioid associated to $\Phi$ is the Grothendieck fibration $\mathbb F_\Phi \to \mathcal D$ associated to the functor $\Phi \rtimes \mathbb N_{\geq 1}: \mathcal D^{op}\to \mathsf{Mon}$ (where $\mathsf{Mon}$ is the category of monoids, regarded as 1-object categories).[2]
If $\Phi: \mathcal D^{op} \to \mathsf{CMon}$ is a divisorial monoid in $\mathcal D$, with group completion $\Phi^{gp}$, and if $0 \to \Phi^{birat} \to \Phi^{gp} \to Pic \to 0$ is a short exact sequence of prehseaves of groups, then there is an action of the presheaf of monoids $\Phi \rtimes \mathbb N_{\geq 1}$ on $Pic$. The presheaf of transport categories for this action has a Grothendieck construction denoted $\mathbb U_\Phi \to \mathcal D$, with canonical map to $\mathbb F_\Phi \to \mathcal D$. This data is called the skeletal Frobenioid of unit-trivial type associated to $(\mathcal D, \Phi, \Phi^{birat})$. Note that the map $\mathbb U_\Phi \to \mathbb F_\Phi$ is a Grothendieck opfibration fibered over $\mathcal D$.
A skeletal pseudo Frobenioid of isotropic type consists of a series of functors $\mathcal C \to \mathbb U_\Phi \to \mathbb F_\Phi \to \mathcal D$ such that $\mathbb U_\Phi \to \mathbb F_\Phi \to \mathcal D$ is a skeletal Frobeniod of unit-trivial type, and additionally $\mathcal C \to \mathcal D$ is a Grothendieck fibration, and $\mathcal C \to \mathbb U_\Phi$ is a Grothndieck opfibration fibered over $\mathcal D$, fibered in groups, such that reindexing along a monoid element $Z \in \Phi(S)$ is always a group isomorphism.
A model Frobenioid is one arising via a similar construction to the construction of $\mathbb U_\Phi$ using only a right-exact sequence $B \to \Phi^{gp} \to Pic \to 0$ of presheaves of abelian groups, where we set $\Phi^{birat}$ to be the image of $B$ in $\Phi^{gp}$. That is, we take $\mathcal C = ``\mathbb U_B"$; the map $\mathcal C \to \mathbb U_\Phi$ comes from the map $B \to \Phi^{birat}$.
So essentially, a Frobenioid is just a way to package the data of a presheaf of exact sequences of abelian groups, plus a bit of positivity data and paying particular attention to the action of $\mathbb N_{\geq 1}$ on everything, into a single category $\mathcal C$. From a categorical perspective, this is certainly something that one is free to do, but there's not really anything forcing one to package things this way. And apparently from the number-theoretic perspective, there's nothing particularly militating for this way of packaging the data. So there doesn't really seem to be anything more to say about them than the definition.
[1] Actually Mochizuki imposes additional conditions on $\mathcal D$ and $\Phi$, but the additional conditions don't seem to play much role in the theory.
[2] Actually Mochizuki requires only that $\Phi$ be a so-called pre-divisorial monoid here, but only the divisorial case plays much role in the theory.