Hi Arthur,
Per my understanding, you might want to customize the People Picker when using the share functionality.
Rather than customizing the OOTB share functionality which would require taking the built-in script and design into consideration, it would be more recommended and
appropriate to create a custom share solution with the custom process of People Picker.
A demo about how to share documents using SharePoint REST API:
http://sharepointfieldnotes.blogspot.com/2014/09/sharing-documents-with-sharepoint-rest.html
About how to create a custom People Picker, client-side People Picker control would be a good choice:
https://msdn.microsoft.com/en-us/library/office/jj713593.aspx
Thanks