Physical: No doubt boxing officers one of the best exercises on the planet in terms of cardio. Much of boxing practice, along with other combat sports, is about conditioning. It is difficult go multiple rounds without having the physical ability capacity to make that happen. Your opponent will simply wear you out and then you are toast!
While much of the time is spent in cardio there is this continuous building of the upper and lower bodies (depending on the sport). I know a semi-professional Muay Thai fighter and I will say that over the course of a year in her conditioning her body has transformed into a lean machine with great upper body strength. You can see the cut in her arms.
We would be remiss if we didn't discuss the increased coordination between different body parts. Your body becomes finely tuned and everything from your toes to your fingertips can coordinate for maximum impact.
Mental: There is something to be said for having the physical confidence of holding your own in any conflict. There really isn't almost any need to use it. Your body language give a lot of signals to others and typically aggressive people simply see you understand your body and know how to use it. We are part animal and our subconscious can pick up on those cues.
That body confidence also influence your mental confidence and self-efficacy. You may be a super smart person but if you also have the ability to use your body you have a well rounded belief in your skills. We should trust ourselves and our instincts. Training against confident people and watching their body movements does make a difference in your situational awareness.
So...I recommend the sport with safety being a primary concern. Either just spar or use head gear. One doesn't need to even hit another person in full contact. It could be muscle memory and coordination of different body parts. The more you get your heart rate up the better off you will be in terms of general fitness.