As much as I am strongly against people accumulating liabilities when they believe within them that they are rich and have arrived, if you are actually wealthy and without debts as you claim, you should be able to afford a car and a house for yourself. No buying any of these does not take away liabilities from you. You will still spend money on renting cars or other means of transportation. Not owning a house does not make you better either, you are still indebted to your landlord.
A combination of both financial discipline and a little comfort and fun is highly adviceable. What is the whole effort for if you can't make life comfortable for yourself?
A house may be a necessity to claim to be rich, but I don't think so for cars, because whether someone is rich or not depends on their surroundings and their own satisfaction. When we have more in anything compared to the people in our surroundings, we can be considered rich (even if we don't have a car). And I think all of that also has to come back to ourselves. When we think that we have enough of everything, then that is enough without needing recognition from others.