This is the first I’ve heard of the effect Mars has on Earth’s Milankovitch cycles (unsurprising, given that the paper is recent and the effect is quite small with a very long period). Earth presumably has a similar effect on Mars, but measuring this would be quite difficult. Keep in mind that we’re able to do this for Earth by analyzing drill cores (that paper uses data from 293 scientific deep-sea drill holes), which we can’t really do for Mars currently. Using other methods, we’ve been able to measure the effects of axial tilt and precession for Mars, but the effect from orbital interactions with Earth would be much more subtle. I’d be surprised if you could find anything on it in the literature.

I also would not expect the Moon to make much of a difference. The Earth-Moon distance is <1% of the Mars-Earth distance even at closest approach, so the Earth-Moon system is essentially a point mass to first order. Additionally, the mass of the moon is ~1% that of the Earth, so the effect there is quite small as well. As I mentioned, measuring Earth-Mars Milankovitch cycles is already difficult for Earth (we apparently only recently did so) while likely infeasible for Mars (currently), and detecting the effects from the Moon would be harder still.