Bujinkan and RMA have a lot in common...on the other end, they have very little.
In Japanese budo, there is a concept called "Shu-Ha-Ri". Essentially, it is learning the form of a technique, changing the form of the technique (henka) and throwing the technique away.
In RMA, it is more "Ha-Ri". There is a principle that isn't engrained in one specific pattern (waza). As the situation changes, even in the beginning stages, the result adapts with what is happening. One could also say that RMA exists plainly in the "Ri" realm.
Bujinkan spends less emphasis on the "Shu" stage of training then most Japanese budo I have encountered....making it closer, in principle, to RMA.
As far as a mental approach, I feel that they are very similar. They both have a "Ninpo" approach to them...and this is found through every stage of training. I often joke with people that I never began to understand Ninpo until a Russian taught it.
If we look at structural similarities, I personally think there are none. There are no ranks in Systema....and even without them, on the whole, people understand their place in things. Everyone in RMA tends to have a beginners approach to training, regardless of years served. In Bujinkan, things aren't quite so clear. People tend to pay attention to a Dan number then what the person has to share**. In Bujinkan...the "beginner's approach" tends to be something sought after as ego takes seat.
**I witnessed at Tai Kai green belts who were sluffed off as training partners by Dan grades from other parts of the country. As I sat back and watched (3 person group...I was watching people around me during my turn out), these two Dan grades *could* have learned a great deal from the green belts....if only their egos and attitudes were left at the door. They had difficulties walking, much less working on what Hatsumi sensei had shown.
One thing that is a great similarity is the idea of using what is at hand. Many times in Bujinkan, techniques are shown with a weapon...and how the technique can remain the same, no matter what is available. This also exists in RMA.