This appears to be a server-side account issue with Apple's iCal services. My wife now has this issue with her account. My account works fine. All settings (aside from the account info) are the same. I obviously am using the software correctly since I have no problems sharing my calendar with her or others. We've tried with different computers and devices, we've tried via Calendar app and via Safari iCloud login... the same issue persists, but only for her. The specified email address/user with which my wife tries to share her calendar is not saved and no invitation is sent. The problem does not occur with my account. Her account will not allow her to share her calendars anymore. This seems like some odd permissions problems with the accounts in question on Apple's calendar server.
This problem is exacerbated since Apple does not allow the deletion of iCloud accounts for various reasons, so if you tied your main business email address (which is printed on all your business stationary, business cards and other collateral) to an iCloud account that later becomes problematic, Apple products can become as dysfunctional as the related account, leading to a sometimes miserable experience across the product range (and I am a die-hard mac user since the SE who is stating this... as well as a long term Apple stock holder.) If Apple continues to tie more and more functionality to the cloud, they need to address these and other related problems which are amplified in such a situation.
I am very concerned with the quality control of Apple software products since Steve Jobs passed away. Software has devolved or become buggier. (Photos is a step backwards from both iPhoto and Aperture. Basic functions, like importation of previously archived Contacts, doesn't always work on 10.11.6 or 10.12.4., etc.)