This is a type of question where there will be many answers. It really depends on a person's perspective.
Most people will say that being rich is a physical trait. In other words, if a person has ten billion dollars, then that person would be considered rich. Even if this person had a poor person's mentality, most people would still consider this person rich. The possession of money is to be rich. What makes this definition attractive is that it is measurable. How much does a person have? Does this person have more than another person?
Another way to answer this question is to say that being rich is a state of mind. For example, if a middle class person is very generous and filled with good works, then we might say that this person has a rich mentality. For most this position will be more attractive, and some will say that being rich is a state of mind.
In my opinion, the truth is somewhere between. For a person to the rich, he or she has to have a certain amount and a mentality of generosity.
Being rich could definitely be measured by both of those, as well as many others. However one would definitely consider the context in which the word "rich", is being used. For example:
If you are in great physical shape, because you exercise daily, you eat healthy, drink plenty of water, and your physical appearance shows that your body mass has only 3% body fat on your entire body, then you are most certainly Rich In a physical context, not to mention that your positive attitude matches your appearance, which could also mean you are rich in that sense.
If you are a person that is full of self confidence, has a really high self esteem, you possess a strong mindset in all areas of thought and actions, and overflowing with fountains of useful information, could certainly describe that you are rich.
Then there is the term that is used when comparing cash dollar values and material possessions such as expensive cars, clothing, jewellery etc. I suppose it may be considered slang when describing financial success and owning an abundance of monetary valueable items and cash itself.
This is a literal term for "rich", every aspect of the word is definitely measured by its surrounding contributing factors, which would measure how it is in fact to be measured.
So to answer your question clearly,
The term "rich" could be measured by one or the other or both, depending on the context you choose to use it in and what other factors are contributing to the context of the word.