What do I say to my husband who insists that his reflection in the mirror is someone else? He calls him his buddy.

Betty Pettit is an elderly woman with advanced Alzheimer's. Her son Joshua Pettit has taken to filming short videos of Betty in her everyday life to bring awareness to AD. Her most popular video is the one where she sees a "woman" in a full length mirror and has no idea who she is:

https://fb.watch/iiM-o0luHN/

See how Joshua handled the situation with his mom, and perhaps you can explain to your husband who the stranger in the mirror is.

Of course, everyone is different. If dh is frightened by the mirror, just cover it up.

I suggest you read this 33 page booklet online, Understanding the Dementia Experience, by Jennifer Ghent Fuller which has some great info about dementia and what to expect from an elder who's been diagnosed with it.

Understanding the Dementia Experience, by Jennifer Ghent-Fuller 
https://www.smashwords.com/books/view/210580

Jennifer is a nurse who worked for many years as an educator and counsellor for people with dementia and their families, as well as others in caring roles. She addresses the emotional and grief issues in the contexts in which they arise for families living with dementia. The reviews for her books are phenomenal b/c they are written in plain English & very easy to read/understand.

Wishing you the best of luck with a difficult situation.

My mother started seeing a little girl. She said that she first saw her in a dream and then she started coming around when she was awake.

I didn’t notice anything different when she was around mirrors. That’s interesting. I wonder how common this is.

If he is calling the image from the mirror his “buddy,” then he doesn’t sound like he is bothered by it.

My mother wasn’t afraid of the “little girl” that she saw. She would tell me about the conversations that she had with this child. She found her to be very comforting.

I just listened. I never told her that there wasn’t anyone else in the room but us.

You can try covering any mirrors that seem to bother him, but most importantly is to just go along with whatever he is saying. If he asks you if the person is still there, you can reply that you're not sure, but if they are you will ask them to leave. And then do so.
When dealing with someone with any of the dementias, one must learn the art of telling little "fiblets," as that will make your life much easier in the long run.
And of course as with all things with dementia, this too shall eventually pass.

Does it bother him? Does he get upset? If not just let him be. He doesn't see himself because himself could be a much younger man in his mind. This man he sees is a stranger.

I had Mom for a while before she went into an AL. I had just helped her in the bathroom. She turned, looked in the mirror and said "Oh My God!" Always wondered if she knew it was her.

Cover mirrors when you have to .
Easy to cover with a towel or you can get a roll of contact paper and cover them that way. Or a bit of acrylic paint, the same color as your wall and paint the mirror. Either the contact paper and the pain are easily scraped off when they are no longer needed.
You can tell him when he is going to the bathroom that the person is not looking.
Don't bother trying to convince him that it is a reflection. the fact that this is another person is his reality.
If he begins to be bothered by "the other person" discuss it with his doctor there are medications that can help with anxiety and agitation.