Is creme brulee supposed to be fatty? I just made creme brulee but it felt a lot like I am eating butter.

I know it should be made with ~35% fat cream so it makes sense to be "fatty". I just have never eaten "real" creme brulee with which to make a comparison and am not sure if it really should taste fatty like butter or if I did something wrong.
No creme brulee should not resemble butter in any way. Not in taste nor in texture. Its consistency should be like that of a thick custard where the eggs determine the firmness of the custard. 
We tend to use single cream with 15% fat for our creme brulee. This works very well and leads to a lighter and less fattier result.