Disallow the "People Search" among general users within Office365 completely.
So far, "People Search" in Delve is referring to the cache information of Exchange contacts. For this reason, the user information will be displayed in Search Result even if it has set to not display in the user selection dialog when sharing on SharePoint Online by settings in advance. Besides, the Search Result of Microsoft Search displays the same result too.
Since we are considering to introduce this product in schools, it is supposed to prohibit User Search between students. Not only the dialog displayed when sharing SharePoint Online but also the feature needs to prevent users from being displayed in search results throughout Office 365 completely.
Gary Huber commented
I do understand the reason this is being asked, but I think a better solution would be to allow the Global Admin to choose if their users can see personal information about each user. In some cases such as mine, we work for a Community Emergency Response Team, it is essential for each user to know how to find everyone's phone numbers and addresses.