The thing about "albums" in the stock Gallery app is that each album is actually a separate directory. Most Gallery apps aren't also file managers, so they don't let you physically move the file from one directory to another. You'd have to use a separate file manager app for that. In addition, most Gallery apps won't recognize nested albums (i.e., a sub-directory of a another album directory) -- they'll just show as a separate album.
If you use Google Photos, the albums are more of a virtual construct -- all of your photos are backed up as one gigantic chronological mass, but you can then create albums and assign photos to them. A single photo can be in multiple albums, without having to duplicate the file. It takes a little getting used to, but it's so much easier to organize photos this way for me.