fedora 23 beta
Gnome 3.18.1
Contacts 3.18.

Currently, there is no way to copy contact information (phone number, email, address, etc.) directly to the contact view, however, you should select the "Edit" option and copy the required information without modifications or save Changes.

The expectation is that the right-click on the required information, a context menu appears and allows you to copy information (and other actions).

For example:

   +56XXXXXXXXX
 ---------^---------
|Copy               |
|Open with other app|
 -------------------

The first choice is obvious, the second opens the app-picker. Both options could be replaced when implemented "share" option (like Android).

Cheers