Kong Operator is installed and upgraded exclusively with the kong/kong-operator Helm chart. This page covers what to know before you upgrade; for the step-by-step procedure, see Upgrade Kong Operator with Helm.
The most important thing to know is that Helm installs Custom Resource Definitions (CRDs) on the first install, but it never updates them afterwards. Any upgrade that ships new CRD fields requires you to apply the CRDs manually before you upgrade the release. Skipping this is the most common cause of a broken upgrade.