TL; DR: These things are sold to hold your phone in the car, but they also work great for keeping parts (including plastic and other non-magnetic items) organized during disassembly.
If you’re taking something apart and there’s any chance you might want to put it back together again, keeping the bits organized is important for a couple reasons—first, you don’t want to lose them onto the floor, under the bench, or into whatever pocket dimension all your missing socks have gone.
Second (and almost as critical when you’re working with modern ultraminiaturized consumer electronics), you don’t want to lose track of how they all went together. Taking notes and/or photos along the way helps a lot with this, but it’s also pretty vital to keep parts from different assemblies and sub-assemblies physically segregated as you go. All those tiny screws may look alike, but they may not be—that one on the interior corner, for example, may be a millimeter or two shorter than the other five, and it may make a big difference if you get ’em mixed up.
A shallow tray works great for the primary function of keeping stuff from getting lost, but one small jostle or bump, and all your carefully-sorted piles are thoroughly scrambled. A tray with compartments is a much better choice, but you may not have enough compartments and those that are available may not be the right sizes, and it will still spill everywhere if you drop it or accidentally slide it off the bench. Magnetic trays are an optimal solution, but only if your parts are made of steel or other ferromagnetic iron, nickel, and/or cobalt alloys. For plastic, aluminum, ceramic, and even stainless-steel parts, they’re no better than a cereal bowl.
Enter the sticky elastomer mat. These things, which are sold to keep your phone and other pocket litter from sliding around in the car, are cheap and widely available in a range of sizes.
They have all/most of the advantages of magnetic trays, except they also work for pretty much all materials. They are permanently adhesive, and though they will pick up dirt, lint, and other debris over time, they can be cleaned and restored to like-new condition with a modicum of effort.