Question for You- Is the calendar, that you are attempting to share with your wife, an iCloud calendar? OR is it a calendar created with some other account such as Yahoo, Gmail etc? This is important to know because if it is NOT an iCloud calendar then the sharing portion is invoked from that service's website.
Now, If it is an iCloud calendar and you do not see the "Share With" option the please check one more thing on both of your iPhones and that is-
Go to Settings-->>iCloud and check to ensure that the Calendar switch is set to ON position. If it is already set to ON then we have something else going on however if it is not then we may have just solved the issue.
Axel F.