socket connection error while building mysql on ubuntu 20.04 docker image

I am trying to build mysql on ubuntu 20.04 docker image, using dockerfile and configuration bash script to install and configure mysql.

Dockerfile:

FROM ubuntu:20.04

RUN apt-get update -qq && apt-get upgrade -y 

ENV MYSQL_DATABASE: 'test'
ENV MYSQL_USER: 'user1'
ENV MYSQL_PASSWORD: 'xxxxx'
ENV MYSQL_ROOT_PASSWORD: 'root'

COPY . .

RUN DEBIAN_FRONTEND=noninteractive TZ=Etc/UTC apt-get -y install tzdata

RUN bash configure_mysql.sh

EXPOSE 3306

Configuration Script:

#!/bin/bash

set -e # Exit script immediately on first error.
#set -x # Print commands and their arguments as they are executed.

# Check if MySQL environment is already installed
RUN_ONCE_FLAG=~/.mysql_env_build_time
MYSQL_PASSWORD="root"

if [ -e "$RUN_ONCE_FLAG" ]; then
  echo "MySQL Server environment is already installed."
  exit 0
fi

# Update Ubuntu package index
apt-get update -y -qq


# Installs MySQL 5.5
echo "mysql-server mysql-server/root_password password $MYSQL_PASSWORD" | debconf-set-selections
echo "mysql-server mysql-server/root_password_again password $MYSQL_PASSWORD" | debconf-set-selections
apt-get install -y mysql-server

# Configures MySQL
sed -i 's/127.0.0.1/0.0.0.0/g' /etc/mysql/my.cnf
sed -i '/\[mysqld\]/a\lower_case_table_names=1' /etc/mysql/my.cnf
echo "MySQL Password set to '${MYSQL_PASSWORD}'. Remember to delete ~/.mysql.passwd" | tee ~/.mysql.passwd; 
mysql -u root -p $MYSQL_PASSWORD -e "GRANT ALL ON *.* TO root@'%' IDENTIFIED BY '$MYSQL_PASSWORD'; FLUSH PRIVILEGES;";
service mysql restart
121.122.1
# Installs basic dependencies
echo "installing unzip, git and curl"
apt-get install -y unzip git curl

# Configures prompt color
sed -i 's/#force_color_prompt=yes/force_color_prompt=yes/g' ~/.bashrc
echo 'source ~/.bashrc' >> ~/.bash_profile
source ~/.bash_profile

# Cleaning unneeded packages
apt-get autoremove -y
apt-get clean

# sets "run once" flag
date > $RUN_ONCE_FLAG

However, the configuration script (line no. 29) returns the following error:

Enter password: ERROR 2002 (HY000): Can't connect to local MySQL server through socket '/var/run/mysqld/mysqld.sock' (2)
The command '/bin/sh -c bash configure_mysql.sh' returned a non-zero code: 1

I have tried changing the permission by passing:

chown mysql:mysql /var/run/mysqld/
chmod -R 777 /var/run/mysqld/

but it doesn't work. May I ask, how can i resolver this issue? Looking forward to the suggestions, Thanks!