Does an action-reaction pair always contain the same kind of force?

We're talking about the "two sides" of the SAME force, so it must be yes, they are of the same type.

For example, if I attract you gravitationally, then you are also attracting me gravitationally. There is only one physical source of the force (in this case, gravity) and it is pulling us both equally and oppositely. Since there's only one thing, it has, and can only have, one type.

Or, to put it another way, according to Newton's third law: When I push against you, you are also pushing me back. It is the same physical thing doing both pushes, in this case electrical repulsion of our atoms.

Or like this: Imaging I support a book of 1N weight in my hand (against earth gravity). I can't support that book without supplying 1N of force to stop it falling. And that force will be provided by the electrical repulsion of the atoms in me vs the atoms in the weight. The book feels that 1N force upwards, and I feel it downwards on my hand. There's only one force doing that. There's ALSO a gravitational attraction pulling the weight towards the earth, balanced exactly by the gravitational attraction pulling the earth towards the weight. So there are (at least) two instances of the third law in play.

If you get down to the fundamental layer, what we perceive as Force on a macroscopic scale is really just exchange of field bosons. So when two particles interact via a boson exchange, it is the same force because it is mediated by the same boson.

The only way you could see asymmetric forces would be if the boson could oscillate into another form. For example, a proton could emit a photon (boson of electromagnetic force), intending to repel another proton electrically, but then the photon could oscillate into a graviton (gravitational force boson) and thus transmit a gravitational force to the second proton.

I've never heard of this happening...

There is no such thing as "the force of A on B." Force is a thing that only exists between objects. That's all there is to it.

An important note is that force is not what physicists consider an observable. That is, you cannot directly measure a force. In classical physics, the only observables are distance, mass, and time. In this context, we infer the force on an object by measuring its mass (observable) and acceleration (length and time observables), and then applying Newton's 2nd law.

Danger: Extremely esoteric stuff below... An interesting philosophical detail here is that we then need to determine mass without using a force (such as gravity, spring, etc.). If we accept the equivalence of gravitational and inertial mass, then we can measure the mass of an object through a collision with another object of known mass.

Addressing comment below

The justification is the following: From a practical point of view it makes total sense to say "the force of A on B." For instance, an engineer might want to know the force a bushing puts on an axle. Or you might want to calculate the torque on a compass needle from an external magnetic field. It would be dumb to ask an engineer to calculate the mutual action between an axle and its bushing. However, the OP asked a deeper, more philosophical question that requires a subtler and more nuanced answer. This requires pulling away the shroud and pointing out that an N3 force pair does not actually consist of two forces. There is ONLY a mutual force. Thus the concept of the equal and opposite forces being of a different nature is nonexistent.