Selected attribute of Drop down list

Your code looks alright. Because your selected output at right place in your screenshot.
I have tried to reproduce the problem but seems all browsers are working correctly.
Maybe you could just create a very simple template without other code to interfere.
Sometimes, a broken HTML structure may also affect.
I tested it both in a WP free html and in my theme by putting it at the bottom of my theme. You may test the following:

1. disable all plugins
2. change to basic theme like 2020 default theme
3. add your code to the bottom of functions.php for a quick test
4. test your page in different browsers
5. you may also print_r() your $selected_division to make sure it is what you want but I think it should be right since the “selected” seems placed in place. Just in case.
6. you may disable browser javascript by extension OR you may test in an incognito mode to avoid interfernce by extensions/addons which could happen sometimes.
7. you may break it part of part like using a static page to make sure it works in a simple form. And then add php to generate and then add JS to control. Doing so could help sorting out the exact problem.

This ensures the theme is free from other interference from plugins or broken structure.

I have tried your code except that I put finite information for test. The selection box works fine in Firefox, Chrome, Safari under MacOS 10.12
For your reference:

<select class="form-control" name="division">
<option value="Select">
    <?php esc_html_e( '-----Select-----', 'donate' ); ?>
</option>
<?php 
     // $selected_division = get_user_meta( $userID, 'donor_division', true ); 

     $selected_division = 'Barisal';

    // just keep your own variables, it is just an example because I have no idea what yours is
    $divisions = array(
        'Barisal',
        'Chattogram',
        'Dhaka',
        'Khulna',
        'Rajshanhi',
        'Rangpur',
        'Sylhet',
    );

    $options="";

    foreach( $divisions as $division ){  

        if( strtolower( $selected_division ) == strtolower( $division ) ){                                                
            $selected = 'selected="selected"';
        }else{
            $selected = '';
        }                                             
        $options .= '<option value="'.esc_attr( $division ).'" '.esc_attr( $selected ).'>'.esc_html( $division ).'</option>';
        }
        echo $options;
    ?>
</select>