Background Items Added

Would suggest the user read from this specific similar question

https://discussions.apple.com/thread/254341579?answerId=258423511022#258423511022

Refer to answers from @etresoft for more details

The it is suggest you remove the "Grammarly, Inc" application.

The communicate with "Grammarly, Inc" regains this behaviour with their Application .

Make sure to mention macOS 13 Ventura and the Notifications issue experienced on this computer

The Notifications of Applications that Run in the Background in Ventura and specific to this version of macOS is a pain in the * * * * * .

The Background Process has always been here in earlier versions of macOS.

It is only now visible in Ventura because Apple Introduced Notification Feature

It was not totally well thought out and Third Party Developers need to work with Apple on this issue

Sorry, the best I can offer would to either take this up with >>

1 - Get Support Choose a product and we’ll find you the best solution.Start now and open an Apple Support Ticket as they are Apple Employees to deal will these types of issues . 

2 - With the Developers of this problematic Application

There is are many similar questions specific to this question

A couple of Links below specific this type of issue

Background Items Added - Apple Community

Background Items Added Problem - Apple Community

Perpetual "Background Items Added" - Apple Community

Am please you received the definitive answers from the " Horses Mouth " aka Apple Support

The Third Party developers have yet to catch-up with all things Apple and specifically Ventura macOS 13

Kind of a shame on them ( Third Parties ).

The Beta versions of Ventura were made available to Developers to Test and Develop their software before it came to production ( Release to Public ) to you and me