Absolutely YES, you can achieve this.
Rule 1: WordPress requires a username. We must provide a username.
Rule 2: Don’t edit WordPress core code.
We can achieve this by hiding username field, get email and store it as username.
Step-1: Remove Username textfield
add_action('login_head', function(){
?>
<style>
#registerform > p:first-child{
display:none;
}
</style>
<script type="text/javascript" src="https://wordpress.stackexchange.com/questions/52302/<?php echo site_url("/wp-includes/js/jquery/jquery.js'); ?>"></script>
<script type="text/javascript">
jQuery(document).ready(function($){
$('#registerform > p:first-child').css('display', 'none');
});
</script>
<?php
});
Step-2: Remove Username error
//Remove error for username, only show error for email only.
add_filter('registration_errors', function($wp_error, $sanitized_user_login, $user_email){
if(isset($wp_error->errors['empty_username'])){
unset($wp_error->errors['empty_username']);
}
if(isset($wp_error->errors['username_exists'])){
unset($wp_error->errors['username_exists']);
}
return $wp_error;
}, 10, 3);
Step-3: Manipulate Background Registration Functionality.
add_action('login_form_register', function(){
if(isset($_POST['user_login']) && isset($_POST['user_email']) && !empty($_POST['user_email'])){
$_POST['user_login'] = $_POST['user_email'];
}
});
Put above code in your theme’s functions.php file.
You can also download a complete plugin to achieve this.