Can't add my wife to Family Sharing Program.

The Apple Watch will be using your Apple ID since it's being paired to your iPhone 6s. It uses the Apple ID of the iPhone it's paired with so no real need for her own Apple ID at this point until she has her own iPhone to pair it with instead.

To do that you need an Watch SE or Series 4, 5 or 6 GPS + Cellular models.

You can set up and manage Apple Watch for someone who doesn’t have their own iPhone—your school-aged child or parent, for example. To do so, you must be the family organizer or parent/guardian in your Family Sharing group.

The iPhone you use to initially pair and set up the Apple Watch must be within normal Bluetooth range (about 33 feet or 10 meters) of the Apple Watch to manage settings and update the software. The person you set up Apple Watch for must be part of your Family Sharing group and have a cellular-capable Apple Watch SE or Apple Watch Series 4 or later. (Your family member’s watch doesn’t have to use the same cellular carrier as the iPhone you manage it with.) See the iPhone User Guide for information about setting up Family Sharing.