Are there any quantities defined analogically to the virtual temperature, but for other gases?
It is true that water is not the only variable constituent of the atmosphere, but it is capable of large variations (0-5%+) that I'm not sure other trace constituents are capable of outside of plumes from smokestacks / cars / factories / etc.  A virtual temperature for other gases would only be meaningful if their variability imparts a significant change in mass in a parcel such that it becomes important for buoyancy.

That being said, you could easily derive such a virtual temperature, as the virtual temperature is just a proxy for density changes and we can easily calculate this.

For each constituent gas, you can use the ideal gas law as

$$p_i = \rho_i R_i T$$

such that each density of gas $i$ is

$$\rho_i = \dfrac{p_i}{R_i T}$$

and then sum these with dry air ($p_d$)

$$\rho = \dfrac{p-\sum p_i}{R_d T} + {\large\sum} \dfrac{p_i}{R_i T} $$

and then solving for $p$ in terms of $\rho$, $R_d$ and collecting the rest of the terms into a variable and calling it $T_v$.

This process could be used to generically define a virtual temperature for any number of trace gases (single or multiple) as long as you know their mixing ratios with dry air.  I'm not aware of anyone using (on Earth) a virtual temp for anything other than water vapor and suspended hydrometeors (e.g. density potential temperature, $\theta_\rho$) but that certainly doesn't mean no one is.