Failed to access keys as result of aws_kms_info ansible module

Any idea how I can access any element from keys list above?

It’s because keys is a method on dict and almost every “object” in ansible/jinja2 is a dict. You can side-step that method resolution via the ["keys"] syntax:

    - debug:
        msg: key_id is {{ vault_aws_kms["keys"][0].key_id }}

the followup question in the comments about the quotes would be

method A:

- set_fact:
    my_key_id: '{{ vault_aws_kms["keys"][0].key_id }}'

method B:

- set_fact:
    my_key_id: "{{ vault_aws_kms['keys'][0].key_id }}"

method C

- set_fact:
    my_key_id: >-
      {{ vault_aws_kms["keys"][0].key_id }}