Mysql Server Not Starting Ubuntu 20.04.3

I have been trying to install the mysql-server on my ubuntu server, but it just gives me this error when I type:

mysql

This is what I get:

ERROR 2002 (HY000): Can't connect to local MySQL server through socket '/var/run/mysqld/mysqld.sock' (2)

If I check if it is on or not with:

sudo systemctl status mysql

I get:

    ● mysql.service - MySQL Community Server
     Loaded: loaded (/lib/systemd/system/mysql.service; disabled; vendor preset: enabled)
     Active: failed (Result: exit-code) since Sat 2022-04-16 23:29:16 UTC; 5min ago
    Process: 474306 ExecStartPre=/usr/share/mysql/mysql-systemd-start pre (code=exited,         status=1/FAILURE)

Apr 16 23:29:16 amir-server systemd[1]: mysql.service: Control process exited, code=exited, status=1/FAILURE
Apr 16 23:29:16 amir-server systemd[1]: mysql.service: Failed with result 'exit-code'.
Apr 16 23:29:16 amir-server systemd[1]: Failed to start MySQL Community Server.
Apr 16 23:29:16 amir-server systemd[1]: mysql.service: Scheduled restart job, restart counter is at 5.
Apr 16 23:29:16 amir-server systemd[1]: Stopped MySQL Community Server.
Apr 16 23:29:16 amir-server systemd[1]: mysql.service: Start request repeated too quickly.
Apr 16 23:29:16 amir-server systemd[1]: mysql.service: Failed with result 'exit-code'.
Apr 16 23:29:16 amir-server systemd[1]: Failed to start MySQL Community Server.

journalctl -xe gives me:

Apr 16 23:38:54 amir-server systemd[1]: Failed to start MySQL Community Server.
-- Subject: A start job for unit mysql.service has failed
-- Defined-By: systemd
-- Support: 
--
-- A start job for unit mysql.service has finished with a failure.
--
-- The job identifier is 2666039 and the job result is failed.

I have tried reinstalling, no luck. What's funny is that the mysql server works on my other linux machine, but I need it to work on this one :(

Any help is greatly appreciated!

[Edit]

pgrep -l mysql

does not return any processes

/var/log/mysql/error.log

does not exist.

And if I do mysqld it gives me:

mysqld: File './binlog.index' not found (OS errno 13 - Permission denied)
2022-04-17T18:30:29.693043Z 0 [Warning] [MY-010101] [Server] Insecure configuration for --secure-file-priv: Location is accessible to all OS users. Consider choosing a different directory.
2022-04-17T18:30:29.710554Z 0 [System] [MY-010116] [Server] /usr/sbin/mysqld (mysqld 8.0.28-0ubuntu0.20.04.3) starting as process 29346
2022-04-17T18:30:29.976620Z 0 [ERROR] [MY-010119] [Server] Aborting
2022-04-17T18:30:30.038737Z 0 [System] [MY-010910] [Server] /usr/sbin/mysqld: Shutdown complete (mysqld 8.0.28-0ubuntu0.20.04.3)  (Ubuntu).

I did what @NioMarvin83 suggested. While installing, I got this:

MySQL has been frozen to prevent damage to your system. Please see /etc/mysql/FROZEN for help.

This is what's inside /etc/mysql/FROZEN:

This MySQL or variant installation has entered "frozen mode". Maintainer
scripts will avoid making changes or starting the daemon until manually
released from this state. See /usr/share/doc/mysql-common/README for
general information about this mode.

In this particular case, an incompatible downgrade attempt has been
detected. This can be resolved in one of two ways:

1. Change the contents of /var/lib/mysql/ to contain database data that
is compatible with the currently installed MySQL or variant daemon
version. For example: you could restore from a backup. Alternatively you
could do a dump using a future version binary and then a restore using
the current version binary.

2. Switch to a MySQL or variant daemon version that is compatible with
the data currently in /var/lib/mysql/. For example, if you have
attempted a downgrade from mysql-server-5.7 to mysql-server-5.6, you
could "apt install mysql-server-5.7" again.

Please resolve this situation and only then remove the /etc/mysql/FROZEN
symlink. You can then run "dpkg-reconfigure <package>" where <package>
should usually be in the form <variant>-server-<version>.