Duplicate Email Notifications

Yes, that’s normal behavior for WP. You can fairly easily suppress all post author notifications with a bit of code, but to selectively only suppress when the author is also the comment moderator/admin it gets a little more complicated. There are various plugins that allow you to further control notifications, you might try out a few of them.
https://www.ads-software.com/plugins/search/comment+notification/

The code to suppress all author notifications:

// prevent post author comment notifications (still allows moderator notifications)
add_filter('notify_post_author', '__return_false');

You can add it to your theme’s functions.php file, but it’ll be lost when the theme updates. To keep it safe, it’s easiest to create a simple, minimalist plugin. You can add any other custom code to this plugin as well.
https://developer.www.ads-software.com/plugins/plugin-basics/#getting-started

I want one to go to the author and one to the admin. Right now they are getting two copies each, of the same email. 4 emails. This can’t be normal..?

I’ve tested this with no plugins, on a post by a normal author and comment by an outside name/email not a member of the site. Admin and author each receive 2 copies of the same message.

Ah, right, you did say 2 each. My bad. No, that’s not normal of course. Since you already tested with no plugins, I suspect your theme. Please switch to one of the twenty* themes and make a test comment.

Do double notices occur with other notifications like password resets? Or just comments?

You are correct, it seems to be related to my theme (Oblivion), and only for comments. I’ll play around with the theme and see if I can hunt down the problem. Thanks for the advice.

You’re welcome. I hope you find it without too much hunting.