What does "not XXX in the sense of it does not XXX" mean?

This could be a weird example, but consider the following statement:
"She is not cute in the sense of being a cat."
Which of the following should this sentence be understood as?
"She is not (cute in the sense of being a cat)." i.e. She is cute, and moreover, not just cute being any random cat–indicating that she stands out to be cute even among her kitty counterparts. In this case, "in the sense of" is only modifying "cute" but not the "NOT".
OR
"She is (not cute) (in the sense of being a cat)." i.e. As a cat, she is NOT cute. Shame on her.

For instance, I see a professor saying this in his handout:
"In fact, the model is not identified in the sense that data cannot distinguish
between model A and B."
In this case he is clearly using "in the sense that" as a "because" clause explaining how this model is "NOT identified".
The fact that there’s another "not" in the subordinate clause just made this sentence even more confusing.
This kind of sentences can really take me a while to understand, which makes me wonder if that’s my fault as a non-native speaker or if there actually exists ambiguity to some extent.

Is this a common way to use "in the sense of/that" to modify the entire negative independent clause?