The Hund's J - Why can this be quantified?

When I first read your question, I found it somewhat puzzling. I have to admit, I still do in part. Why? Well, even your link defines $J$ as a sum of matrix elements $langle m,m'|V_{ee}|m',mrangle$. Mathematically, each of these matrix elements is an integral involving wave functions in the system. If we know these wave functions or know how to approximate them it is conceptually straight-forward to calculate the integrals, save for possible divergences. Leaving such issues aside, I'd say $J$ is eminently quantifiable, at least in principle. (And demonstrably also in practice for e.g. few-body systems and atoms.)

Now, I suspect the above is a kind of pedantic answer that doesn't really get to the heart of the question you wanted to ask. In the question itself and your comments below it, you make a clear distinction between the Hubbard $U$ and Hund's $J$ by saying one is linked to on-site electron-electron interactions and calling the other a pure exchange effect. You then ask how $J$ can be given a value similar to that of the Coulomb repulsion. This suggests that there might be a misconception about the nature of exchange effects at play. In fact, Hund's type exchange while magnetic in effect, is due to a combination of the Coulomb force and the Pauli exclusion principle.

A simple version

The simplest version of this physics shows up already in the treatment of two-electron atoms (notably Helium). This problem is discussed in textbooks like Sakurai's Modern Quantum Mechanics in more detail than I will go here. Thus I'll just jump to the result for the energy associated with the Coulomb repulsion between the two electrons (which is treated as a perturbation on top of a non-interacting ground state): $$ leftlangle frac{e^2}{r_{12}} rightrangle = I pm J, tag{1} $$ where $r_{12}$ is the distance between the two electrons. The upper sign goes with a spin singlet state, and the minus sign with a spin triplet state. $I$ and $J$ are called the direct and exchange integrals, given by $$ I = int mathrm{d}^3 mathbf{x}_1 int mathrm{d}^3 mathbf{x}_2 | psi_{100} left( mathbf{x}_1 right)|^2 | psi_{nlm} left( mathbf{x}right)|^2 frac{e^2}{r_{12}},tag{2} $$ $$ J = int mathrm{d}^3 mathbf{x}_1 int mathrm{d}^3 mathbf{x}_2 psi_{100} left( mathbf{x}_1 right) psi_{nlm} left( mathbf{x}_2right) frac{e^2}{r_{12}} psi_{100}^star left( mathbf{x}_2 right) psi_{nlm}^star left( mathbf{x}_1right).tag{3} $$ I don't want to go into detail about the notation here, but the difference in structure is clear. In $I$ we have a density-density interaction, but $J$ involves a mixing or exchange of the two wave functions at the two coordinates. While the sign of $pm J$ is determined by the spin state and enforced by the antisymmetry of the wave function, the magnitude $J$ is set by the Coulomb repulsion, and is indeed quantifiable.

In materials

I will here loosely follow the notation of another book I like: Physics of Transition Metal Oxides by Maekawa et al. For concreteness we consider two $3d$ elecrons, labeled $1$ and $2$, with positions $r_1$, $r_2$ and wave functions $Psi(r_1)$, $Psi(r_2)$. In the context of this stack we can think of them either as two localized molecular orbitals (as in LCAO) or as two Wannier functions. The Coulomb interaction between them is $$ H = frac{1}{2} int int Psi^dagger (r_1) Psi^dagger (r_2) frac{e^2}{|r_1-r_2|} Psi(r_2) Psi(r_1) mathrm{d}tau_1tau_2,tag{4} $$ where we integrate over spatial variables and sum over spin degrees of freedom. (Both steps are hidden inside the $tau$ symbols.) We can now decompose the wave function $$ Psi (r) = sum_{n,m,sigma} c_{nmsigma} psi_{nm} (r) theta_sigma,tag{5} $$ where $n$ denotes site, $m$ denotes orbital quantum number, and $sigma$ the spin quantum number. $psi_{nm}$ is the spatial wave function, and $theta_sigma$ is the spin wave function. In second quantization, $c_{nmsigma}$ is the annihilation operator associated with an electron with these quantum numbers. Then, if we include the spatial integration inside the matrix elements, we can write $$ H = frac{1}{2} sum_{n,m,sigma} langle n_1 m_1, n_2m_2| frac{e^2}{|r_1-r_2|} | n_3m_3,n_4m_4rangle c^dagger_{n_1 m_1 sigma_1} c^dagger_{n_2 m_2 sigma_2} c_{n_4 m_4 sigma_2} c_{n_3 m_3 sigma_1}.tag{6} $$

This expression includes multiple processes. First, we have the one-site ($n_1=n_2=n_3=n_4=n$) one-orbital ($m_1=m_2=m_3=m_4$) case,, $$ H_U = frac{1}{2}sum_sigma langle n,n| frac{e^2}{|r_1-r_2|} | n,nrangle c^dagger_{nsigma}c^dagger_{n,-sigma} c_{n,-sigma} c_{nsigma} = Un_{nuparrow}n_{ndownarrow},tag{7} $$ with $n_{nsigma}=c_{nsigma}^dagger c_{nsigma}$. This is of course nothing but our friend the Hubbard interaction, where the messy details of the matrix element is hidden behind the symbol $U$.

Two electron on the same site ($n_1=n_2=n_3=n_4=n$), but in different orbitals produce the so-called Hund coupling (c.f. Hund's rules). We get begin{align} H_H = frac{1}{2} langle m_1,m_2 | frac{e^2}{|r_1-r_2|} |m_1,m_2rangle sum_{sigma_1,sigma_2} c_{m_1sigma_1}^dagger c_{m_1sigma_1}c_{m_2sigma_2}^dagger c_{m_2sigma_2} tag{8}\ - frac{1}{2} langle m_1,m_2 | frac{e^2}{|r_1-r_2|} |m_2,m_1rangle sum_{sigma_1,sigma_2} c_{m_1sigma_1}^dagger c_{m_1sigma_2}c_{m_2sigma_2}^dagger c_{m_2sigma_1}tag{9}\ equiv K_{m_1m_2} n_{n_1}n_{n_2} - 2J_{m_1m_2} left( mathbf{S}_{m_1m_2}cdot mathbf{S}_{n_2} + frac{1}{4} n_{m_1}m_{n_2}right).tag{10} end{align} Here $K$ is a on-site inter-orbital Coulomb density-density interaction, and $J$ represents the Hund's coupling. You can see that this matches (up to coefficients) Eq. (11) in the notes you linked in the question. Note that the magnitude of $J$, like that of $U$, is related to the strength of the Coulomb force between the two electrons.

Similarly, inter-site terms like direct exchange can be derived, and ultimately quantified.

First-principles calculation

Although the Hubbard $U$ and Hund $J$ are often treated as semi-empirical parameters in materials modeling, and sometimes assigned values using rules of thumb (that I don't know the origin of), there are some first-principles frameworks that can be employed to calculate them accurately. To get reliable values, screening effects must be taken into account. This can be done while downfolding to effective Hamiltonians using e.g. the constrained Random Phase Approximation (cRPA) and constrained GW approximation, which can use DFT band structures as inputs (though there are subtleties concerning double-counting of correlation effects). For details on cRPA see e.g. Chapter 7 in this set of lecture notes. For cGW, some developments are described in this and this paper.

It's worth noting that such calculations often uncover significant non-local (i.e. intersite) Coulomb interactions, see e.g. this paper on $alpha$-RuCl$_3$ from last year and references therein. Such non-local interactions are often overlooked, but might have important effects in strongly correlated systems.