It is known that fluid drag and Brownian motion form an example of the fluctuation-dissipation theorem (FDT). But there are different types of drags. Are all of them dissipative or not? What is the relevant type of drag in terms of the FDT relation?
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Yes, whenever you couple your system to a thermal bath or apply some kind of coarse graining, you must have a some kind of drag. You can see the FDT in two ways:
If you include some noise in your description and you are at equilibrium (the system is not driven and has an Hamiltonian structure), then, automatically, for self-consistency, you must include a drag. Otherwise, the steady-state probability distribution would not be equilibrium like. For example, you would obtain long-ranged correlations without a drag.
If you have some kind of drag or more generally dissipation and, let's say a potential landscape, any physical dynamics would just make you perform a gradient descent (dissipate the energy). To consistently incorporate thermal fluctuation on top of this gradient descent, you simply have to add a noise with well chosen correlation, such that you are sampling an equilibrium distribution.
Now to your question, yes, any kind of drag works. At least if your system is thermal; in principle even systems with dry friction for example should have some kind of FDT, in practice, the energy scales at which you see dry frictions are too high to be affected by any meaningful thermal fluctuations. We call these systems athermal systems. But, for example, Navier-Stokes equation, for consistency, must include a noise:
$$ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \eta \nabla^2 \mathbf{u} + \nabla \cdot \boldsymbol{\Sigma}^{\rm random} $$
where $\boldsymbol \Sigma$ is a random stress with correlation: $$ \langle \Sigma_{ij}(\mathbf{x}, t) \rangle = 0 $$
$$ \langle \Sigma_{ij}(\mathbf{x}, t) \Sigma_{kl}(\mathbf{x}', t') \rangle = 2 k_B T \eta \left( \delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} \right) \delta(\mathbf{x} - \mathbf{x}') \delta(t - t') $$ The correlations of the noise respect the FDT with respect to the viscosity $\eta$, which is a drag. However, it is a very peculiar drag, it acts only on small wavelength (otherwise the gradient is small). And, the FDT requires that the random stress also only acts on small scales, hence the $\nabla$. Another way to see this is that, momentum must be locally conserved, hence any term that you add to your equation of motion must be as a gradient of something. Of course, this is equivalent to seeing this new term as a mesoscopic noise arising from the discrete collisions taking place in the fluid; which must conserve momentum. If you allow for a compressible fluid, you also have an equation for the temperature of the fluid, which also incorporates a random noise. These equations were first written down by Landau and Lifshitz and are called fluctuating hydrodynamics equations. They are very important because they show for example that 2D hydrodynamics is somewhat pathological due to a diverging renormalization of the transport coefficients in the infinite system size limit (Is the Navier-Stokes equation valid in $d=2$ spatial dimensions?).
What about simpler? Let's take a diffusion equation:
$$\partial_t\rho=D\boldsymbol \nabla^2\rho + \rm Noise$$
According to what I told you above, what should be the noise here? It must respect the conservation of density, so somehow, it should be written, again, as a local divergence of a current, and indeed we find:
$$\partial t\rho = D\boldsymbol \nabla\cdot\left(\boldsymbol \nabla \rho + \boldsymbol J^{\rm random}\right)$$
I let you find the autocorrelation of $\boldsymbol J^{\rm random}$. In a more complex system, with maybe phase separation, you would have:
$$\partial t\rho = D\boldsymbol \nabla\cdot\left(\boldsymbol \nabla \dfrac{\delta F}{\delta \rho} + \boldsymbol J^{\rm random}\right)$$
with $F$ a Landau Free energy. To learn more on these, you might be interested in the Theory of dynamic critical phenomena
Edit:
You might be interested by this paper https://www.sciencedirect.com/science/article/pii/S1631070517300695:
We shall focus mostly on questions related to the modeling of physical phenomena where the discreteness of matter is the source of fluctuations. A good example of questions posed by the extension of the Langevin theory is the case where the friction is not viscous friction, namely not proportional to the velocity. This arises when one considers the Brownian motion of a particle of mesoscopic size diffusing on a solid surface whilst remaining close to it. Then the viscous drag proportional to the velocity is usually replaced by a friction law where the drag is not proportional to the velocity. This leads to major theoretical problems. A model of solid friction often used in this field amounts to add to the viscous drag $-gamma u$ a Coulomb friction, which is a constant times a discontinuous and strongly nonlinear function of the velocity, something like $-\nu u/|u|$. As shown [13] by Goldenfeld and collaborators (see also references [14] and [15] for the Langevin equation with Coulombic solid friction), the equilibrium distribution function of the velocity fluctuations for such a friction law with the usual white noise term is not Maxwellian. This forbids us to consider this model as a fair description of the equilibrium Brownian motion of a particle on a solid surface, but does not exclude at all that it is relevant for other situations. The requirement that the equilibrium fluctuations of the velocity have a Maxwellian distribution is central to a picture of equilibrium. Similar questions (yet unsolved) arise when dealing with the thermal fluctuations in electric circuits including a diode leaving the current to flow in one direction only. In such cases, presumably, the splitting into a viscous “friction” and a random Langevin-like force statistically independent does not hold anymore. It should also be stressed that adding a Langevin-like force with vanishing mean value to non-linear equations is not consistent in the sense that taking then the average, one does not recover the original equations because of the induced correlations.
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