Your passwords for the database does not match.
But the error is about it not being able to reach the database on the hostname.
I can run it with this compose file:

services:
  jellystat-db:
    image: postgres:16-alpine
    container_name: jellystat-db
    restart: unless-stopped
    environment:
      POSTGRES_USER: ${POSTGRES_USER}
      POSTGRES_PASSWORD: ${POSTGRES_PASSWORD}
    volumes:
      - postgres-data:/var/lib/postgresql/data
    networks:
      - jellystat
  jellystat:
    image: cyfershepard/jellystat:latest
    container_name: jellystat
    restart: unless-stopped
    environment:
      POSTGRES_USER: ${POSTGRES_USER}
      POSTGRES_PASSWORD: ${POSTGRES_PASSWORD}
      POSTGRES_IP: jellystat-db
      POSTGRES_PORT: 5432
      JWT_SECRET: ${JWT_SECRET}
      TZ: Europe/Paris # timezone (ex: Europe/Paris)
      JS_BASE_URL: /
    volumes:
      - jellystat-backup-data:/app/backend/backup-data
    depends_on:
      - jellystat-db
    networks:
      - traefik
      - jellystat
    labels:
      - traefik.enable=true
      - traefik.docker.network=traefik
      - traefik.http.routers.jellystat.entrypoints=https
      - traefik.http.routers.jellystat.rule=Host(`${HOSTNAME}`)
      - traefik.http.routers.jellystat.tls.certresolver=http
      - traefik.http.routers.jellystat.service=jellystat
      - traefik.http.services.jellystat.loadbalancer.server.port=3000
      - traefik.http.services.jellystat.loadbalancer.server.scheme=http
networks:
  jellystat: {}
  traefik:
    external: true
volumes:
  postgres-data: null
  jellystat-backup-data: null