This is colloquial speech, so don't read too much into the root meaning of the words.
"Insane" does literally infer some kind of mental illness, but over time it has become used as a superlative, that is to exaggerate something.
I can't say for certain, but I believe the root of this change in meaning may stem from use of terms like "insanely jealous". Jealousy, while not a healthy emotion, is certainly not a mental illness. Saying someone is insanely jealous does not mean they have actually lost their sanity, but simply that their jealousy is so strong it has affected their rational thought. But over time, idioms can take on a meaning of their own as people pick up what they think it means and use it without considering the meaning of the individual words. Nowadays it is quite common for people to use "insane" or "insanely" with other words to exaggerate them, or take something to the furthest reach of the imagination.
Other words are similarly used (misused?) this way - "ridiculous" for example. That literally means something is deserving derision or mockery, but you will hear people say "a ridiculous amount" of something, to mean more than is reasonable. Something that has stuck in my mind was back in 2002 when I watched a behind-the-scenes documentary about the movie Spiderman in which young actor James Franco described older co-star Willem Dafoe as being "in ridiculous shape". Taken literally that would mean his shape was something laughably strange! But of course he meant that his shape, or physical health, was ridiculously good - or so much better than you would imagine for someone of his age. Just as he used "ridiculous" without any other adjective to describe something, you will also hear people say "that's insane!" when they think something is beyond what is expected.
Saying someone "cares an insane amount" doesn't make a whole lot of sense literally, and even used as hyperbole (which it clearly is) I would call it an example of language being poorly understood by the speaker. Still, language evolves and is alive. Young people speak this way, and you will hear expressions like this. I take it to mean that the person cares much more than you would expect.