Have you considered using the POSTGRES_DB environment variable ?

services:
  db:
    container_name: postgres
    image: postgres
    environment:
      POSTGRES_USER: ${POSTGRES_USER}
      POSTGRES_PASSWORD: ${POSTGRES_PASSWORD}
      PGDATA: /data/postgres
      POSTGRES_DB: bank
    volumes:
      - db:/data/postgres
    ports:
      - "5332:5432"
    networks:
      - db
    restart: unless-stopped
    healthcheck:
      test: [ "CMD-SHELL", "pg_isready -d postgres" ]
      interval: 30s
      timeout: 10s
      retries: 5
networks:
  db:
    driver: bridge

volumes:
  db:

First, use alpine images if possible.

Second, use ENV variables for prepare database configuration. More info.

Third, use /docker-entrypoint-initdb.d directory for extend image. More info.

Example stack with devlopment environment.

version: '3.2'
services:
  example:
    image: "${EXAMPLE_SERVICE}:${EXAMPLE_SERVICE__VERSION}-development"
    build:
      context: .
      dockerfile: Dockerfile.development
    networks:
      - net
    ports:
      - 5000:5000
    depends_on:
      - example__migrator
    stdin_open: true
    tty: true
    environment:
      POSTGRES_PASSWORD: example
      POSTGRES_DB: example
      POSTGRES_USER: example
      POSTGRES_HOST: pg.example_net
      POSTGRES_PORT: 5432
      FLASK_APP: app.py
      FLASK_DEBUG: 1
    volumes:
      - .:/home/example
  example__migrator:
    image: "${EXAMPLE_SERVICE}:${EXAMPLE_SERVICE__VERSION}-development"
    build:
      context: .
    command: sh -c './wait-for pg.example_net:5432 -- python manage.py db upgrade'
    networks:
      - net
    depends_on:
      - pg
    environment:
      POSTGRES_PASSWORD: example
      POSTGRES_DB: example
      POSTGRES_USER: example
      POSTGRES_HOST: pg.example_net
      POSTGRES_PORT: 5432
  pg:
    image: postgres:10.0-alpine
    networks:
      - net
    ports:
      - 5432:5432
    volumes:
      - pg_data:/var/lib/postgresql/data/pg_data
    environment:
      POSTGRES_PASSWORD: example
      POSTGRES_DB: example
      POSTGRES_USER: example
      POSTGRES_HOST: pg.example_net
      POSTGRES_PORT: 5432
      PGDATA: /var/lib/postgresql/data/pg_data
networks:
  net:
volumes:
  pg_data: