Firstly, from the middle paragraph it seems you imply that adding, subtracting or changing variables of potentials stands in literal connection to energy transfers and but that's not the case. You can write down all quantities for all situations and in any case, there is only one and the same second law of thermodynamics at work. For any system parameter which is a priori left open, that law tells the system which configuration must be chosen. The change of parameters is the consequence of this and then, as a consequence of this, the energy in the system is changed. That's the math of thermodynamics in its beautiful abstract form.
Now you say
In every other case, we exchange heat, volume, and particles with the reservoir. How do we justify writing $G=F−HM$. Though it is true that H is maintained constant, we don't exchange magnetization with a "magnetic reservoir".
and what you do is put the focus on particular applications of the theory. In particular, you put emphasis on some parameters (volume and particle number) which are, in some models, conserved in total. However, thermodynamics only wants the total energy you set up to be constant. What the first and second law do is make the system pull energy until the maximal entropy state is reached and, in your case here, that energy is stored in form of magnetization.
Likely, there is no reservoir of vibrational excitation of $\mathrm N\mathrm H_3$, but you can still transfer translational energy to vibrational energy. Your analogies lack anyway. There are many field theories with non-conserved particle number and if the system is free to produce them, it will do. And I'd also not say that "volume is exchanged" if you do chemical reactions out in the wild. You have you vessel with volume $V_1$ and when your system blows up, due to exoterthermic reactions say, would you say you "exchange" volume with rest of the world volume $V_2=\infty$? What really is at work in the $P$-$V$ case is the process towards a force-equlibirum, bound again by energy conservation, here manifest in Newtons third. The vessel-volume $V_1$ pushes against gravitational/atmospheric force until it doesn't, and then equlibirum is reached. The energy contribution for ideal gas is $PV$. Likely, magnetizable systems work with or against the magnetic field until they have enough.
PS: I ranted about the different potentials and the Legendre transformation in thermodynamics here.