Not just because you might get it back if you lose it, but also to prevent a full-blown SAR (Search and Rescue).
Make sure to use a permanent marker (silver Sharpie works well). Motion Board Shop offers free Lost & Found stickers with reward.
It is also wise to take pictures of your gear and write down serial numbers if you need to identify it later (due to theft, loss, etc.)
From U.S Coast Guard in the Pacific Northwest, their Paddle Smart program.