Wife's Apple Watch displays husband's name and activity data

Your wife should erase the watch and pair it again, setting it up as a new device, using her Apple Account.

Unpair and erase your Apple Watch - Apple Support

Also have a look at this excellent User Tip by Lawrence Finch:

How to “unshare” an Apple ID - Apple Community

If you use Family Sharing, all members of the Family Group will be able to download and use supported apps on your iOS and watchOS devices:

How to set up Family Sharing on your iPhone, iPad, or Mac - Apple Support

Make sure that each family member has a unique Apple Account - Apple Support

The Fitness app is tied to the Apple ID utilized in the App Store for App Store purchases. To change that log out of the App Store and log back in with the Apple ID you want to appear in the Fitness app on the device(s). I suspect that she has her own Apple ID for almost everything but for the App Store account her device(s) are tied to your Apple ID. This is related to her use of your Apple ID for App Store purchases. You may have inadvertently or intentionally set it up this way.

Jeff627 wrote:

I appreciate your fast response, but it leads to more confusion for me. Why would the Fitness app use the ID used to log in to the app store

The Fitness App appears in the App Store so apparently Apple decided to tie it to the Apple ID used to get things from the App Store. When two Apple IDs are utilized on a device, such as a Mac, iPhone, Watch, etc. and one is for the device and the other for purchases/downloads, then you can see this bifurcated situation.

The Health app also appears in the App Store (Apple Health) and it has a section on the Summary page (that one can access via the picture or symbol in the upper right for the user) that lists "Devices." You might inspect what is there on your wife's iPhone to try to sort this out. There is also a place to set up "Sharing" in the Health app but I am not familiar with how it works; you should check your settings for that also.

and why use it to announce Good Morning on the watch?

That seems to be associated with the Fitness or Health app and their identifiers. Apple makes it possible to use two Apple IDs on one device but recommends against it, perhaps to avoid these sorts of confusions.

Why not use the ID from the watch itself? Seems counter-intuitive to me.

I agree. But I can't change how Apple has set it up.

But I can change the user on the app store. Just trying to reduce charges.

As I mentioned earlier, Apple makes it possible to use two Apple IDs on one device but recommends against it, perhaps to avoid these sorts of confusions.