The answer at that link is incorrect.
The “Edit Animals” text from their answer uses the edit_item label, not singular_name. There’s a much longer list of labels that you’ll need to provide when registering a post type than just those two. The full list is kept here.
As for singular_name, as far as I can tell, this is only used to label menu items and link suggestions:
In both those examples “Page” is where the singular_name is used.