Can we use quantities other than temperature to describe thermal equilibrium?

In general, thermal equilibrium means maximizing the entropy. The reason we use temperature is that very often, two systems can do this by exchanging energy. Under an exchange of energy $dE$, $$dS_{text{tot}} = frac{dS_1}{dE_1} , dE + frac{dS_2}{dE_2} , (-dE) $$ so the maximum entropy is achieved when this is zero, and the systems have the same $$frac{dS}{dE} = frac{1}{T}$$ where this is really a definition of $T$.

In general, you can exchange other things too. For example, if a container is separated in two by a movable piston, then the total volume of the two pieces is conserved, and we can maximize entropy by exchanging volume. Then in thermal equilibrium, they have the same $$frac{partial S}{partial E} bigg|_V = frac{1}{T}, quad frac{partial S}{partial V} bigg|_{E} = frac{p}{T}$$ where the second equation serves as the thermodynamic definition of pressure. If the total number of some kind of particle is conserved, and the systems can exchange particles, we equalize $$frac{partial S}{partial E} bigg|_{V, N} = frac{1}{T}, quad frac{partial S}{partial V} bigg|_{E, N} = frac{p}{T}, quad frac{partial S}{partial N} bigg|_{V, E} = - frac{mu}{T}$$ where the third equation defines the chemical potential. If there were $n$ separate types of such particles, we'd have $n$ separate chemical potentials that would be set equal.

There are plenty of more exotic options too. In general, there is a potential for every conserved quantity which is conserved, can be exchanged between the systems, and affects the entropy in the thermodynamic limit. (On the other hand, in an introductory course it's reasonable to focus on systems with only one or two, to avoid too much complication with partial derivatives.)