The Apple TV box is old but I have done a full update.
What is the exact model?
Identify your Apple TV model - Apple Support
Find your Apple TV model number - Apple Support
What is its version system software?
Apple TV does not give me the option to choose my favorites album
I don’t think Favorites is an actual album, in the way that can be selected as album on Apple TV. I think you’ll need a custom named manual album for that.
only play stream
My Photo Stream is a ‘smart’ (dynamic) album of recent iCloud synced photos for the last 30 days. If you created 16 photos in those recent days, that that is what will be in it. Links to older photos will be removed automatically, to keep it actually recent.
Try creating a new special iCloud album with the images that you would like for the screen saver. Then you can select that on Apple TV. Organize this on iPhone, iPad, Mac, or web, whichever is most comfortable for you.
If needed, you’ll find more expertise on Photos and iCloud Photos in the Photos forum.