Yes, that's correct. I'm not a developer for Corsair but I feel like I have a pretty general understanding of how the software works. Essentially, you load a default profile onto the keyboard using the Corsair software (CUE). It will revert to that profile whenever it is not receiving instructions from CUE (if the computer is off or disconnected). Once the keyboard starts receiving instructions from the Corsair software (when the computer boots and loads CUE), it will automatically switch to that profile.
So if Computer A is running Profile A, when you switch to Computer A, the keyboard will switch to Profile A
Similarly, Computer B is running Profile B. Switching your KVM to Computer B would cause the keyboard to run Profile B
If either computer is off in this situation, the keyboard would switch to its default profile because it would no longer be receiving instructions from CUE. However, in my experience, the keyboard won't be animated unless it's connected to a computer that is running and has the CUE program running.
Addendum: Also, in my experience, the keyboard animation stops at the user login screen. Not sure if this only happens on Windows 10 or if earlier versions of Windows cause the same behavior. My guess is that the login screen invokes some sort of kernel-level code to prevent malicious applications from taking control of the login screen. Regardless, they keyboard won't load the profile from CUE until after you log in. Until then, you'd be stuck with the keyboard's default profile.