Pagination in single.php

Custom loops do not produce pagination like that.

You have two options:

  1. Use custom archive page and provide custom php template for that (i.e. archive-my_name.php), in this custom template you could provide all other necessary content. Then you could use standard built-in pagination functions https://developer.wordpress.org/themes/functionality/pagination/
  2. Continue in the direction you started, with single.php, but you would need to provide your own pagination, together with rewrites for custom pages. Again, https://developer.wordpress.org/themes/functionality/pagination/#methods-for-displaying-pagination-links provides starting point for that.

I advise using option 1. as easier.