You won't be able to tell the difference once they are stained.

My first thought was that if you want mahogany color---use mahogany. The color will be much better. Maple is difficult to stain/dye evenly. It is also difficult to get maple as dark as most mahogany.

Birch is very bland, very little grain differences. I do not know how well it takes stain/dye, however.

Pricing will vary with area and type of wood----there are basically two different types of maple(hard and soft). Soft is less expensive, but more difficult to work. Hard comes in several grades, #2 grade has some color differences---dark wood in the light. #3 has more dark.

Much of the cabinets will probably be made using plywood with the wood of choice veneer. Veneered ply often takes stains/dyes differently than solid wood of the same species.

My advice is to find a cabinet maker you like and have that company show you what they can do to achieve your choices. That will give you the opportunity to make specific decisions.

Birch stained with "dark red mahogany" aniline dye is a perfect substitute for mahogany. But real mahogany isn't going to be a preemptive premium in cost from birch.
Casey