Yes, Judy, it's not uncommon to find this; during construction this is a quick method to cover the stair stringers, and often it's never traded out for new material.
There are some overlay wood floors available which are only 3/8" to 1/2" thick, interlocking tongue and groove at the joints, and work nicely on stairs.
To install this type of material, I would first remove the existing scrap wood and replace it with a 1" floor decking material, then your new finish wood.
There's a few considerations here to keep in mind, particularly your riser heights. Your new finished stairs should wind up having as near to the same height for each riser as they have now, or you'll have uneven drops at the top and bottom of the staircase.