Hi there, this DIYer came up with a creative solution that might inspire you. https://www.hometalk.com/35576054/thrift-store-repurposed-frames
Take a wide piece of wood (or plywood) and build up higher levels one-by-one by gluing or nailing pieces that are more narrow on top of each other. When you have built it up to the height you want drill holes and glue dowels in the holes. The dowels must be of a size that fits into the bobbins and thread spools. Paint or stain it in a color of your choice.
Keep in mind that thread, on spools or bobbins, should never be stored where it will receive much sunlight or bulb light, as it will fade the colors, and break down the fibres. This will result in distorting the tension, more knotting, more fraying, and more breakage.
Good quality thread is not cheap.
If you do not have the floor space for shallow drawers, or if you want your thread 'on display' such as in the example above, then at least cover it out of the light when not in use.