There are several procedures that people call "Stabilized". The best one is complete impregnation with mono-acrylic resins which are then polymerized.The wood is harder, and polishes to a bright shine, without finish. Some other procedures are soaking with waterproofing and similar chemicals/resins. Thompson's, Nelsonite, Minwax wood hardener....etc.
To check if the wood is stabilized, sand the end grain of a block or scale to 400 grit. Wet half of the sanded area. Wipe off the water. If the wet area is darker looking than the dry area it is not completely stabilized. This test shows if the water is sucking up into the grain.
Some woods that are stabilized by various processes will pass this test, but will still feel soft and light weight. If it won't absorb water, then it should be OK.
A coating of some sort of finish won't hurt, but it should not be sucking into the wood if the wood is stabilized. I like thin CA for those woods that need a little more surface sealing and shine. There is nothing wrong with the natural flat look and feel of wood either. No rule says it has to shine.
Stacy