get_template_part() based on get_post_type() for a custom post type instead of get_post_format()

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Make sure you are using the correct template file:

First, you need to make sure index.php is being used as the template. You may install What The File Plugin, which will show what template file is being used at the top bar when you’re logged in.

Set the proper condition for portfolio:

Once you are sure that index.php is used as the template and you have template-parts/content-portfolio file within your active theme, then you may use the following CODE for the get_post_type() check to determine which template-part should be used:

<?php /* Start the Loop */ ?>
<?php while ( have_posts() ) : the_post(); ?>

    <?php
        if( get_post_type() === 'portfolio' ) :
              get_template_part( 'template-parts/content', get_post_type() );
        else :
              get_template_part( 'template-parts/content', get_post_format() );
        endif;
    ?>

<?php endwhile; ?>

Update: CODE swap based on comment:

Replace:

<?php
    if ( has_post_format( array( 'gallery', 'video', 'image' ) ) ) {
        get_template_part( 'template-parts/content', get_post_format() );
    }
    else { 
        get_template_part( 'template-parts/content', 'single' );
    }
?>

with:

<?php
    if( get_post_type() === 'portfolio' ) {
        get_template_part( 'template-parts/content', get_post_type() );
    }
    else if ( has_post_format( array( 'gallery', 'video', 'image' ) ) ) {
        get_template_part( 'template-parts/content', get_post_format() );
    }
    else { 
        get_template_part( 'template-parts/content', 'single' );
    }
?>