Is “the loop” a template tag?

The Template tag is just a name chosen for a variety of function. It doesn’t really matter what you call them, and they are functions.

However, the while( ) loop itself is a simple PHP feature. It uses the have_posts method of the WP_Query() class to detect whether there is any posts to display or not. The main query itself is already set when you load a page, which can be accessed via global $wp_query;.

So, putting all together, this can be an opinion based question. The WP_Query() class doesn’t exists in the File section of Template Tags page, so I would call it a “No”.