It means + present simple

why the present simple is used in the first example

It is present simple because it is a habitual action, as in I wake up at 7:15.

Alarm sound can't mean the fact that I wake up. I think that would be "My cat learned that alarm sound means I will wake up right after it started sounding"

Yes, your interpretation that I will wake up right after it started sounding is correct. The original sentence is a simplified, condensed form of the longer sentence. You might argue that the original sentence is overly simplified and condensed to the point of not being logically valid.

But the meaning is still clear.

People will use simple sentences because they aren't native speakers, or because they are children, or because they are being lazy, or because they want to abbreviate the idea into a bite-sized form. There are many reasons. Even if logic hasn't been strictly followed, if the meaning is sufficiently conveyed then sometimes it's enough.

This example with the cat also favors such a format. Because it's trying to paint a picture that the cat has simple thoughts, but they are just advanced enough to understand that "the alarm" = "human wakes up".

Here are some suggestions for a more complete sentence:

My cat has learned that the sound of the alarm means I'm about to get up.
My cat has learned that the sound of the alarm means I'm about to wake up.