How can you identify if something is made of mahogany?

Firstly, find out if it is softwood or hardwood. To do that, press your fingernail on the wood in some hidden place (you do not want to damage the piece). If your fingernail leaves a mark, it is softwood. If it does not, then it is hardwood. Mahogany is a hardwood.

Next, find out if it is solid wood or veneer (a thin layer attached to the surface of a less expensive wood). You can do that by looking at the corners. If you can see the edge of a thin layer, you are looking at a veneer.

If it passed the first two tests, look at the colour and pattern of the grain. Mahogany has a light brown colour if not stained but looks deep reddish brown when varnished, wet or stained. It takes a stain very well and there won't be a lot of contrast. The grain is fine (oak, by comparison, has a wide grain) and is long, luxurious and can have a wave to it.

Looking close up, you will see grain is made up of many many dark fine lines, from 1/16th of an inch to about an inch.

Of course, a picture is worth a thousand Words, so look online for pictures of mahogany and compare it to cherry, walnut, oak, maple and pine.