I found the issue: as I'm running on OpenShift 3.11 (Kubernetes 1.11), the API config had to be changed so it supports admission controllers.
MutatingAdmissionWebhook:
configuration:
apiVersion: v1
disable: false
kind: DefaultAdmissionConfig
ValidatingAdmissionWebhook:
configuration:
apiVersion: v1
disable: false
kind: DefaultAdmissionConfig
This block must be present in the master-config.yml in the section admissionConfig.pluginConfig. After restarting the apiserver, the webhook started to kick in. But the sidecar was still not injected, because of some permission issues. Granting the consumer app's service account cluster-admin permissions or access to the privileged SCC (equivalent of PSP) helped, but then also introduces other security issues.