Клиент не видит схему бд H2 на сервере

Все оказалось гораздо банальнее:
1. Используйте одну строку подключения: jdbc:h2:tcp://localhost/~/h2databases/my_serverdb
2. Разберитесь с настройками: private String settings = ";CIPHER=AES;AUTO_SERVER=TRUE;MULTI_THREADED=1"; и аргументами сервера, так как убрав их все заработало.
3. Я надеюсь, что в рабочем коде вы закрываете подключения.

Пару советов:
4. У Вас немного нарушена логика:
- Запуск сервера - это запуск сервера и создание таблиц ни как не связано с ним
- Вынесите создание таблиц и пользователей отдельно
- Не делайте запуск и работу с сервером в конструкторе

Итоговый код, который у меня заработал, хоть и вырвиглаз ))
Сервер:

import java.sql.Connection;
import java.sql.DriverManager;
import java.sql.ResultSet;
import java.sql.SQLException;
import java.sql.Statement;
import java.util.logging.Level;
import java.util.concurrent.CountDownLatch;
import java.util.logging.Logger;
import org.h2.tools.Server;
/**
 *
 * @author ezhov_da
 */
public class H2DbTest
{
    private static final Logger LOG = Logger.getLogger(H2DbTest.class.getName());
    private String path = "jdbc:h2:tcp://localhost/~/h2databases/my_serverdb";
    private String settings = ";CIPHER=AES;AUTO_SERVER=TRUE;MULTI_THREADED=1";
    private String user = "serveradmin";
    private String password = "filepassword adminpassword";
    private Server server;
    private CountDownLatch countDownLatch = new CountDownLatch(1);
    public H2DbTest()
    {
        Thread thread = new Thread()
        {
            @Override
            public void run()
            {
                try
                {
                    System.out.println(":: H2 server side ::");
                    // starting the TCP Server
                    server = Server
                            .createTcpServer(/*
                             * "-tcpAllowOthers",
                             * "-webAllowOthers",
                             * "-pgAllowOthers" */).start();
                    countDownLatch.countDown();
                } catch (SQLException ex)
                {
                    Logger.getLogger(H2DbTest.class.getName()).log(Level.SEVERE, null, ex);
                }
            }
        };
        thread.start();
        try
        {
            countDownLatch.await();
        } catch (InterruptedException ex)
        {
            Logger.getLogger(H2DbTest.class.getName()).log(Level.SEVERE, null, ex);
        }
        try
        {
            Class.forName("org.h2.Driver");
        } catch (ClassNotFoundException e)
        {
            System.out.println(e);
        }
        Statement statement = null;
        ResultSet resultSet = null;
        String sql = null;
        try (Connection connection = DriverManager.getConnection(path/* + settings */, user, password);)
        {
            statement = connection.createStatement();
            System.out.println("Port: " + server.getPort());
            System.out.println("URL: " + server.getURL());
            System.out.println("Status: " + server.getStatus());
            sql = "CREATE TABLE IF NOT EXISTS MYUSERS("
                    + "userid int, username varchar(255),"
                    + "PRIMARY KEY(userid) )";
            statement.execute(sql);
            sql = "INSERT INTO MYUSERS (userid,username) VALUES (1,'denis')";
            statement.execute(sql);
            sql = "INSERT INTO MYUSERS (userid,username) VALUES (2,'fil')";
            statement.execute(sql);
            System.out.println(connection.getCatalog());
            resultSet = statement.executeQuery("SELECT userid, username FROM MYUSERS");
            while (resultSet.next())
            {
                String userid = resultSet.getString("userid");
                String username = resultSet.getString("username");
                System.out.println(userid + " " + username);
            }
            /*
             * DatabaseMetaData meta = connection.getMetaData();
             * ResultSet rs = meta.getTables("./serverdata", null,"%", null);
             * System.out.println("tables list:");
             * while (rs.next())
             * {
             * System.out.println(rs.getString(3));
             * } */
            // makes new user
            sql = "CREATE USER MARY PASSWORD 'Marypassword'";
            statement.execute(sql);
            // gives mary rights only to read
            sql = "GRANT SELECT ON MYUSERS TO MARY";
            // sql = "GRANT ALTER ANY SCHEMA TO MARY";
            System.out.println(
                    statement.execute(sql)
            );
        } catch (Exception e)
        {
            LOG.log(Level.ALL, "f", e);
            System.out.println(e.getMessage());
        }
        while (true)
        {
        }
    }
    public static void main(String[] args)
    {
        H2DbTest h2DbTest = new H2DbTest();
    }
}

Клиент:

import java.sql.Connection;
import java.sql.DriverManager;
import java.sql.ResultSet;
import java.sql.Statement;
import java.util.logging.Level;
import java.util.logging.Logger;
/**
 *
 * @author ezhov_da
 */
public class Client
{
    private static final Logger LOG = Logger.getLogger(Client.class.getName());
    private String path = "jdbc:h2:tcp://localhost/~/h2databases/my_serverdb";
    private String user = "MARY";
    private String password = "Marypassword";
    public Client()
    {
        try
        {
            Class.forName("org.h2.Driver");
        } catch (ClassNotFoundException e)
        {
            System.out.println(e);
        }
        System.out.println(":: H2 client side ::");
        Statement statement = null;
        ResultSet resultSet = null;
        String sql = null;
        try (Connection connection = DriverManager.getConnection(path, user, password);)
        {
            statement = connection.createStatement();
            System.out.println(connection.getCatalog());
            resultSet = statement.executeQuery("SELECT userid, username FROM MYUSERS");
            while (resultSet.next())
            {
                String userid = resultSet.getString("userid");
                String username = resultSet.getString("username");
                System.out.println(userid + " " + username);
            }
        } catch (Exception e)
        {
            LOG.log(Level.ALL, "f", e);
            System.out.println(e.getMessage());
        }
    }
    public static void main(String[] args)
    {
        Client client = new Client();
    }
}