I solved in this way:
I logged in with root username
mysql -u root -p -h localhost
I created a new user with
CREATE USER 'francesco'@'localhost' IDENTIFIED BY 'some_pass';
then I created the database
CREATE DATABASE shop;
I granted privileges for new user for this database
GRANT ALL PRIVILEGES ON shop.* TO 'francesco'@'localhost';
Then I logged out root and logged in new user
quit;
mysql -u francesco -p -h localhost
I rebuilt my database using a script
source shop.sql;
And that's it.. Now from php works without problems with the call
$conn = new mysqli("localhost", "francesco", "some_pass", "shop");
Thanks to all for your time :)