MySQL8.4からmysql_native_passwordがデフォルト無効になったらしい。
以下の小細工でmysql_native_password有効なMySQLが立ち上がる。

docker-compose.yml
docker-entrypoint-initdb.d/
  001_setup.sql

docker-compose.yml

services:
  mysql:
    image: mysql:8.4.2
    container_name: mysql
    command: --mysql-native-password=ON
    ports:
      - 3306:3306
    environment:
      MYSQL_ROOT_PASSWORD: password
      MYSQL_DATABASE: testdb
    volumes:
      - ./docker-entrypoint-initdb.d:/docker-entrypoint-initdb.d

docker-entrypoint-initdb.d/001_setup.sql

ALTER USER `root`@`%` IDENTIFIED WITH mysql_native_password BY 'password';
ALTER USER root@localhost IDENTIFIED WITH mysql_native_password BY 'password';

# docker compose up -d
// ここでdocker-entrypoint-initdb.d/が実行される時間がしばらくあるので待つ20秒くらい
# docker compose exec mysql mysql -ppassword -e "select user, host, plugin from mysql.user;"
+------------------+-----------+-----------------------+
| user             | host      | plugin                |
+------------------+-----------+-----------------------+
| root             | %         | mysql_native_password |
| mysql.infoschema | localhost | caching_sha2_password |
| mysql.session    | localhost | caching_sha2_password |
| mysql.sys        | localhost | caching_sha2_password |
| root             | localhost | mysql_native_password |
+------------------+-----------+-----------------------+