Favicons are linked to each theme. They are stored in each theme’s image folder, and referenced in each themes header.php file. All current themes use .png files, not .ico.
The webtrees favicon should no longer be there, and will not be in the next version.
Browsers in general stopped expecting favicons to either be called that, or use the .ico format, or be placed in a system’s root folder many years ago (except IE of course which took a lot longer to wake up) Most will still default back to those if nothing better exists though.
To be perfectly correct, your header .php file should probably have at least 14 “favicons’ of different sizes referenced and available, to cater for desktop, IOS devices of various sizes, android devices, and Win8 ‘panels’. But as kiwitrees / webtrees doesn’t perform particularly well on small devices I have never considered it necessary to include all of them in every theme.
However, I have recently been using http://realfavicongenerator.net/ for some WordPress sites, so may add a set to kiwitrees themes sometime. That will create all the necessary images and related HTML in one go from a suitable single image file.