Can't connect to MySQL via my application

By default MySQL is configured to allow connections to 127.0.0.1 (localhost). If you try to connect to it via the public IP, even within the Linode, MySQL won't like that. Here's a quick test I made on a fresh Wordpress Linode:

Using the public IP:

>>> cnx = mysql.connector.connect(user='root', password='random', host='173.230.157.121', database='wordpressdb')
Traceback (most recent call last):
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/connection_cext.py", line 232, in _open_connection
    self._cmysql.connect(**cnx_kwargs)
_mysql_connector.MySQLInterfaceError: Can't connect to MySQL server on '173.230.157.121:3306' (111)

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/__init__.py", line 272, in connect
    return CMySQLConnection(*args, **kwargs)
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/connection_cext.py", line 85, in __init__
    self.connect(**kwargs)
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/abstracts.py", line 1003, in connect
    self._open_connection()
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/connection_cext.py", line 235, in _open_connection
    sqlstate=exc.sqlstate)
mysql.connector.errors.DatabaseError: 2003 (HY000): Can't connect to MySQL server on '173.230.157.121:3306' (111)

Using localhost:

>>> cnx = mysql.connector.connect(user='root', password='random', host='127.0.0.1', database='wordpressdb')
Traceback (most recent call last):
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/connection_cext.py", line 232, in _open_connection
    self._cmysql.connect(**cnx_kwargs)
_mysql_connector.MySQLInterfaceError: Access denied for user 'root'@'localhost'

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/__init__.py", line 272, in connect
    return CMySQLConnection(*args, **kwargs)
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/connection_cext.py", line 85, in __init__
    self.connect(**kwargs)
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/abstracts.py", line 1003, in connect
    self._open_connection()
  File "/usr/local/lib/python3.7/dist-packages/mysql/connector/connection_cext.py", line 235, in _open_connection
    sqlstate=exc.sqlstate)
mysql.connector.errors.ProgrammingError: 1698 (28000): Access denied for user 'root'@'localhost'

Notice the errors are different. When using the public IP, I get the same error as you did. When using the localhost IP (you can use the string "localhost" as well instead of the IP, by the way), MySQL lets me in but rejects my attempt because the password is wrong.