Display different content on homepage depending on post type

Question

You are doing it wrong on many levels here:

You should not be using a custom query in place of the main query. Rather use pre_get_posts to alter the main query accordingly. Go back to the main loop and add the following in your functions file or in a custom plugin

add_action( 'pre_get_posts', function ( $q )
{
    if (    $q->is_home()
         && $q->is_main_query()
    ) {
        $q->set( 'post_type',      ['photo', 'video', 'web'] );
        $q->set( 'posts_per_page', -1                        );
    }
});

is_singular() is the wrong check here. is_singular() checks if you are on a singular post page, so it will always return false on any page that is not a singular post page.

What you need is get_post_type() to get the post type the post belongs to and test against that

if ( 'photo' === get_post_type() ) :
   echo "<a href="https://wordpress.stackexchange.com/photo">photo</a>";

elseif ( 'video' === get_post_type() ) :
    echo "<a href="http://wordpress.stackexchange.com/video">video</a>";

elseif ( 'web' === get_post_type() ) :
    echo "<a href="http://wordpress.stackexchange.com/web">web</a>";
 endif;

Lastly, if you want to link to the archive pages of those post types, you should be using get_post_type_archive(). Something like the following would work
```
<a href="https://wordpress.stackexchange.com/questions/220376/<?php echo get_post_type_archive_link("Photo' ); ?>">Photo</a>
```
instead of echo "<a href="https://wordpress.stackexchange.com/photo">photo</a>"

EDIT

From comments, I have done an extensive answer on why not to use custom queries in place of the main query. Feel free to check it out here. It should answer quite a few questions for you