why can't i add a contact as an emergency contact

When you attempt to add an emergency contact in the Health app, note the fine print under the addition:

"Your emergency contacts will receive a message saying that you have called emergency services when you use Emergency SOS. Your current location will be included in these messages."

Undoubtedly, your wife does not have a telephone number in her contact information that is capable of receiving such a message.

This same thing happened to me. Oddly, my husband's phone number was only in my Favorites and for some odd reason, not in my contacts. Only his email addresses were in my contacts. Once I discovered this, I added him directly to my Contacts app (I thought I had done this years ago!) with his phone numbers, saved it, went back into the Health app and I could then add him. Very strange, but it worked. A contact will be greyed out when it only shows an email for that contact.

Easy workaround if you don't want to unlink contacts. Just create a separate new contact called 'wife' and then her name and put in the same telephone no. And then choose that as emergency contact

Undoubtedly, you are 100% wrong. Apple Health Kit is not parsing contact data properly for linked contacts. I had to unlink (and as a result duplicate) my girlfriend's contact info to add her. Not only is this feature buggy, the text you quoted is as unhelpful as your response.

Do you have telephone number contact for her in your Contacts app? The only reason that I have just seen when checking my device is that the person only has an email address in the contacts, or no phone number.

I had the same issue when trying to add my dad. I went to my contact list and saw that he was showing up as a contact with yahoo & icloud. I deleted his contact info for yahoo (which was only an email address) and then it let me add him. I might have been able to just add his number to the yahoo contact option but I didn't see any point in having him under both options. Might be something worth looking at. Hope this helps.