The expression ***in the sense that***, implies that from the existing senses, the speaker is concerned with a particular one. So it's restrictive. And it can be negated without ambiguity. But I agree that if another negation follows ***in the sense that***, the sentence can become intricate.

Your funny example,
>She is ***not cute** in the sense* of being a cat.

means that

> She is ***cute, but not** in the sense* of being a cat [i.e. not because she is a cat, not due to her "catness", it is not her being a cat that makes her cute - maybe she has a certain affinity to emotions or gestures that seem human, or something else, we don't know]

So the sentence asserts a quality, but denies a particular reason for that quality.
> A is not B in the sense that C (A is B, but not because of C).

"Not" negates the verb "is" but not in an absolute way, as the NP "in the sense of being a cat" comes to limit this negation. The negation is true ***only in this sense...***

Your other sentence

> In fact, the model ***is not identified** in the sense that data **cannot distinguish*** between model A and B.

means that ***the identity*** of the model ***does not lie in the inability of the data*** to act in a certain way. The model is not identified by a certain fact *[that of the data being unable to distinguish between A and B]*, and that fact happens to be expressed by a negative clause. So the model **can** be identified, ***but not*** by means of that negative fact.