I realized there were a few mistakes here after posting this question...

1. I used PostgreSQL 11 for replication prior so I assumed they worked the same way (which of course is wrong, there are some changes). The recovery.conf is omitted from PostgreSQL 12 and it gave this error message FATAL: XX000: using recovery command file "recovery.conf" is not supported when I had it. so I had to remove it from my ConfigMap.

2. I confused about Docker's Entrypoint & Command to Kubernetes' Command & Args. After being corrected by my senior that Kubernetes Command will override the Docker Entrypoint, I'm going to need and use only Args afterwards.

The following are the changes I made to my ConfigMap and Deployment.

apiVersion: v1
kind: ConfigMap
metadata:
  name: pg-replica
  labels:
    app: postgres
    name: pg-replica
data:
  POSTGRES_DB: postgres
  POSTGRES_USER: postgres
  POSTGRES_PASSWORD: mypassword
  pg_hba.conf: |
    # Contents
  postgresql.conf: |
    data_directory = '/var/lib/postgresql/data'
    # the contents from recovery.conf are integrated into postgresql.conf
    primary_conninfo = # host address and authentication credentials
    promote_trigger_file = # trigger file path
  extra.sh: |
    #!/bin/sh
    postgres -D /var/lib/postgresql
---
apiVersion: v1
kind: Service
metadata:
  name: pg-replica
  labels:
    app: postgres
    name: pg-replica
spec:
  type: NodePort
  ports:
  - nodePort: 31000
    port: 5432
  selector:
    app: postgres
    name: pg-replica
---
apiVersion: apps/v1
kind: Deployment
metadata:
  name: pg-replica
spec:
  selector:
    matchLabels:
      app: postgres
      name: pg-replica
  replicas: 1
  template:
    metadata:
      labels:
        app: postgres
        name: pg-replica
    spec:
      containers:
        - name: pg-replica
          image: postgres:latest
          imagePullPolicy: "IfNotPresent"
          ports:
            - containerPort: 5432
          envFrom:
            - configMapRef:
                name: pg-replica
          volumeMounts:
            - name: pg-replica
              mountPath: /var/lib/postgresql/data
            - name: replica-config
              mountPath: /var/lib/postgresql/postgresql.conf
              subPath: postgresql.conf
            - name: replica-config
              mountPath: /var/lib/postgresql/pg_hba.conf
              subPath: pg_hba.conf
            - name: replica-config
              mountPath: /docker-entrypoint-initdb.d/extra.sh
              subPath: extra.sh
          args:
            - "-c"
            - "config_file=/var/lib/postgresql/postgresql.conf"
            - "-c"
            - "hba_file=/var/lib/postgresql/pg_hba.conf"
      volumes:
        - name: pg-replica
          persistentVolumeClaim:
            claimName: pv-replica-claim
        - name: replica-config
          configMap:
            name: pg-replica

The arguments in Args will set the location of the .conf files to where I specified.

For further steps in Replication:

1. After the pod is up, I manually ran the pod's shell with kubectl exec.

2. I removed all the files from the data-directory for step 3 (to copy files from master pod).

rm -rf /var/lib/postgresql/data/*

3. Use pg_basebackup to backup data from master node.

pg_basebackup -h <host IP> --port=<port number used> -D  /var/lib/postgresql/data -P -U replica -R -X stream

And that's it. Now I managed to have my pg-replica pod replicating my master pod.

You can use Kubernetes postStart lifecycle hook to create your schema on given Postgres DB. Just update your deployment file with the lifecycle block.

apiVersion: v1
kind: Pod
metadata:
  creationTimestamp: null
  labels:
    run: postgres
  name: postgres
spec:
  containers:
  - image: postgres:10.4
    name: postgres
    resources: {}
    lifecycle:
      postStart:
        exec:
          command: ["/bin/bash","-c","sleep 20 && PGPASSWORD=$POSTGRES_PASSWORD psql $POSTGRES_DB -U $POSTGRES_USER -c \'CREATE SCHEMA IF NOT EXISTS key;\'"]
    envFrom:
    - configMapRef:
        name: postgres-config
    volumeMounts:
    - mountPath: /var/lib/postgresql/data
      name: data
  dnsPolicy: ClusterFirst
  restartPolicy: Never
  volumes:
  - name: data
    emptyDir: {}
status: {}