Though I only did Aikido for a short period, I was much more attached to the philosophical and meditative side rather than martial. Actually some of the ukemi wrist locks and finger injuries were a bit concerning so I gave it up in favor of drumming because I realized really can't do both at the level I want and continued the mental/spiritual aspect independently. Eventually the joy of music superseded any martial aspirations.
It has arguably been one of biggest influence both in my life and drumming, I don't claim to have mastered anything (just trying). But to be literal, purposefully breathing is fundamental for a relax attentiveness which relates musical awareness in an improv setting, improved sense of groove and just empathetic listening, with a point of mental focus that both physical and conceptual (usually several cm below the navel is what I've learned). Consequently it helps with speed and endurance as well. It also involves adapting to your situation to make the music happen whatever the scenario. Posture and balance was also eye opening for center of gravity, it helps my feet in the sense I never want my balance of my body by my legs at all. Making use of centripetal force whenever possible, this comes up Aikido/weapons all the time and certainly in stick work which has been talk about extensively. Also never having a favorite right/left....maintaining equal balance to take out weaknesses which is why I forced to learn both right and left lead playing. Balance just isn't physical but also musically; fill and making space along with harmony and dissonance...the dance of extremes together. But above all, if you want to be better at anything (fill the blank: drumming, martial arts...whatever), you have to work on being a better person. There is no mistake that some of the musical greats (or the least one I admire) happen to be super nice people and there is a level of spiritual training to develop this fully (I mean this on an agnostic level). And if you are a jerk or immature, the music has a way of showing just that.
There is ton of rhythm within martial arts but even that is limited... If you want to go further, there is the concept of rhythm in the universe and how it may relate with ki/qi/chi or whatever your want to call it which I feel rhythm is the same. Philosophically, you can start to contemplate why is rhythm something we are even celebrating...it is fundamentally very primitive to human existence; from our first heartbeat to uncontrollable joy of dance or even sex. Then there are the vibrations of waves and particles that created just this sentence, then expand that to various universes....all doing their own single stroke rolls at different rates and we have a human sense to capture some of this reality and have a conscience awareness that allows us to enjoy it...for some reason! Nonetheless, there is alot of philosophy in martial arts and which is completely intertwined with music/drumming on this level as well.