Trying to add a photo from my gallery/library to my favorites album, however, all other albums show and are an option, but the favorites album is not there therefore not an option.

Just to add to what Old Toad said: Those "albums" at the top of the sidebar under "Photos" aren't really albums like the ones you make with "File>New Album." Those things, like Recents, People, Places, Favorites, etc, are special pre-set ways to view your pictures-- you don't add pictures to any of them. Favorites is a view that shows all the pictures that you've identified as favorites with the little heart tag. "Recents" shows pictures in the order your pictures were added to the Library-- not necessarily in the order they were taken. Places lets you look at your pictures according to where the GPS (or your manual entry) says the pictures were taken. You can't change how these views work.

But you can put some pictures in an Album and sort them as you like. That's what the "add" button is for. You can also make Smart Albums that will include your favorites from 2022, for instance.

There is an "Image>Add to Favorites" that sets the little heart tag for you so that the picture shows up in Favorites and any smart album that incudes favorites.

You don't use the menu "Add to" to add a photo to the Favorites album. While viewing the thumbnail move your cursor to the lower left hand corner of the image unto the heart appears. Click on it so it becomes filled and it's in the Favorites album:

I think, the question is about Photos iOS, on an iPhone, not a Mac, given the screenshots, Richard. There is still no way to create our own smart albums on an iPhone or iPad (I wish it were), but what you wrote will apply to the Favorites album, Recents, and Imports as well on iOS devices.