What can I line a collectible card box with that is not felt or flocking?
You could take inspiration from old suitcases: these used either fabric (woven linen or cotton), or paper. (I actually suspect that they used bookbinding paper/cloth, more on that later.)

Reasonably sturdy paper would be easiest to handle - patterned wrapping paper from a craft store, maybe, or even a piece of wallpaper.

Fabric can be tricky (stretching, glue seeping), but a smooth, woven upholstery fabric is a reasonable option.

If you can get your hands on book cloth, that's definitely a great choice. It's fabric lined with paper, sold in sheets or rolls. Takes glue well, doesn't stretch or wrinkle, easy to cut in straight lines. The downside is that it can be expensive or hard to find, depending on your location.