"In the sense that/of" is typically used to clarify the meaning of something one is trying to express. Example: That woman is hot in the sense that she as a fever, not that I think she's attractive. You use "sense that" before a secondary clause ("she has a fever" in my example), and "sense of" when it's used before a verb ending in "ing". (example: "When I said his nose is running, I used the word 'running' in the sense of dripping, not of moving quickly.")

It's an idiom for explaining something, like "I wouldn't call myself Jacob, in the sense that Jacob isn't my real name" Stupid example but it gets the idea across.