Multivariate Cryptography: Security of the affine transform T
Almost a year since I posted this question, back then I was starting the study of the field. Now I can answer myself:

The security of both affine transforms $(T,S)$ relies on the IP2 assumption defined in the Isomorphism of Polynomials. Plus, if the private polynomial $\mathcal{F}$ is known there are other implications, like finding $T$ or $S$ would reveal the other affine transform.

This is, any multivariate scheme where $F,S$ or $F,T$ is known gives us the ability to break it. When $F$ is not known, as usually, we could obtain $T\cdot S$ as a matrix if both are linear transform instead of affine, and the linear part of $F$ is the identity matrix. From here nothing else can extracted, from the moment, in my study.

Regarding IP1, it can be broken when $F$ is known, or when $F$ is unknown by quering an oracle $n$ times with the canonical vectors $\overrightarrow{e_i}$.

Besides other techniques to strip out $T,S$ exist, i.e: when the characteristic is not $2$ and both transforms are linear.

There is more work not published in literature that I've discovered, i.e: the special structure of matrices $S$ that make the private polynomial invariant in a way that we can revover $T$, and more techniques that the audience must await for.