Note that this issue is related to behavior mentioned in #1846.

When one uses "host-less URI" as specified on the JDBC support page, e.g.: jdbc:tc:mysql:5.7.12:///mydatabase, DB name is also ignored and a default test name is used instead.
This code fragment from org.testcontainers.jdbc.ConnectionUrl#parseUrl method clearly shows how it is handled:

//In case it matches to the default pattern
Matcher dbInstanceMatcher = Patterns.DB_INSTANCE_MATCHING_PATTERN.matcher(dbHostString);
if (dbInstanceMatcher.matches()) {
	databaseHost = Optional.of(dbInstanceMatcher.group(1));
        databasePort = Optional.ofNullable(dbInstanceMatcher.group(3)).map(value -> Integer.valueOf(value));
        databaseName = Optional.of(dbInstanceMatcher.group(4));
}

then org.testcontainers.containers.JdbcDatabaseContainerProvider#newInstanceFromConnectionUrl defaults DB name to test:

final String databaseName = connectionUrl.getDatabaseName().orElse("test");

In my particular case an init SQL script, which starts with the following instruction, fails with Caused by: java.sql.SQLSyntaxErrorException: Access denied for user 'test'@'%' to 'mydatabase' error:

CREATE DATABASE IF NOT EXISTS `mydatabase`;

Aforementioned documentation page says:

Note that the hostname, port and database name will be ignored; you can leave these as-is or set them to any value.

which apparently is not the case and /// syntax, which includes DB name only, is recommended and multiple examples are provided.

I suggest to improve "host-less" URI handling by honoring DB name and clarify the documentation on this matter. Will make a PR for this if project owners are OK with such resolution.