The new sharing system for Power Apps is very similar to the one used for SharePoint:
This system will not allow the pasting of users in bulk in the 'Add a name, group,,,,' box. I've tried with different browsers and separators and using names vs emails, to no avail. Fortunately, the option to share using "Classic sharing" is still available:
Let's hope Microsoft will not take that option away, or they will allow for users to be added in bulk in the new sharing window!