Update
Based on the update to your question, you're trying to run the mysqladmin ping command inside the container. mysqladmin is attempting to connect as the root user, but authenticating to your database server requires a password.
You can provide a password to mysqladmin by:
- Using the 
-p command line option 
- Using the 
MYSQL_PWD environment variable 
- Creating a credentials file
 
If we move the root password out of your image, and instead set it at runtime, we can write your docker-compose.yml file like this:
version: "3"
services:
  mariadb:
    restart: always
    image: mariadb_image
    container_name: mariadb_container
    build: topcat_mariadb/.
    environment:
      - "MYSQL_ROOT_PASSWORD=$MYSQL_ROOT_PASSWORD"
      - "MYSQL_PWD=$MYSQL_ROOT_PASSWORD"
    healthcheck:
      test: ["CMD-SHELL", 'mysqladmin ping']
      interval: 10s
      timeout: 2s
      retries: 10
And then in our .env file we can set:
MYSQL_ROOT_PASSWORD=pw1
Now after the container starts up we see that the container is
healthy:
$ docker ps
CONTAINER ID   IMAGE            COMMAND                  CREATED          STATUS                    PORTS      NAMES
c1c9c9f787e6   mariadb_image    "docker-entrypoint.s…"   28 seconds ago   Up 27 seconds (healthy)   3306/tcp   mariadb_container
As a side note, it's not clear from this example why you're bothering to build a custom image: it's better to set the environment variables at runtime, rather than creating an image with "baked-in" credentials.
Previous answer
I can't reproduce your problem when using the mysql client or Python code. Given the following docker-compose.yml:
version: "3"
services:
  mariadb:
    restart: always
    image: mariadb_image
    container_name: mariadb_container
    build: topcat_mariadb/.
  shell:
    image: mariadb:10.6.4
    command: sleep inf
(The directory topcat_mariadb contains the Dockerfile from your
question.)
If I exec into the shell container:
docker-compose exec shell bash
And run mysql like this:
mysql -h mariadb_container -u user1 -p db1
It works just fine:
root@4fad8e8435df:/# mysql -h mariadb_container -u user1 -p db1
Enter password:
Welcome to the MariaDB monitor.  Commands end with ; or \g.
Your MariaDB connection id is 6
Server version: 10.6.4-MariaDB-1:10.6.4+maria~focal mariadb.org binary distribution
Copyright (c) 2000, 2018, Oracle, MariaDB Corporation Ab and others.
Type 'help;' or '\h' for help. Type '\c' to clear the current input statement.
MariaDB [db1]>
It looks like you may be using sqlalchemy. If I add a Python container
to the mix:
version: "3"
services:
  mariadb:
    restart: always
    image: mariadb_image
    container_name: mariadb_container
    build: topcat_mariadb/.
  shell:
    image: mariadb:10.6.4
    command: sleep inf
  python:
    image: python:3.9
    command: sleep inf
And then run the following Python code in the python container:
>>> import sqlalchemy
>>> e = sqlalchemy.engine.create_engine('mysql+pymysql://user1:user1pw@mariadb_container:3306/db1')
>>> res = e.execute('select 1')
>>> res.fetchall()
[(1,)]
It also seems to work without a problem.