According to the screen shot of the connection to the DB server you have MariaDB and you have the MariaDB client installed on the Web server.
You may not have setup users with access rights priv's on the DB server. The prompt shown in the clip of the terminal session with the remote DB server indicates you have connected to it as root user ... probably the only user configured at present.
Did you create a database for Moodle on the remote DB server? If NOT do so, installation via Web will not create a database for you ... you must do that manually ON the db server.
After having done the above, test by changing the db uer to root user ... with no password.
That, BTW, will throw a warning if you ever do a health check after you get Moodle installed.
See the Moodle docs for setting up the moodle DB and the user to be used to interact with the moodle DB.
'spirit of sharing', Ken