So from what I was understanding, a customer-managed CMK means that it is a CMK with imported key material.
Imported key material cannot be automatically rotated, as stated in AWS documentation: https://docs.aws.amazon.com/kms/latest/developerguide/importing-keys.html#importing-keys-considerations
However there is this page which states that customer-managed CMKs can be automatically rotated every 365 days: https://docs.aws.amazon.com/kms/latest/developerguide/rotate-keys.html
So then, what is a customer-managed CMK? Does it not mean that you have a CMK with imported key material?
When it comes to customer-managed CMKs, you have the option to either choose an external KeyMaterial or Key Material generated by KMS.
I believe the automatic rotation (optional) is available for symmetric customer managed CMKs that use AWS KMS.
Please refer to the following FAQs page for more information. (espl. the question Why should I create my own customer master keys?)