Oak, ash, beech, maple, black locust...all work for axe handles with American eyes. But there are various barriers that can prevent folks from making their own handles:
•Access to wood--Getting your hands on the wood requires either land, connections, or a lumber supplier in your area that can get you the quantity and quality you need (usually small) at a price you can afford. Some folks have access to none of these.
•Access to tools--If working direct from the tree, you'll need to split billets or have access to a mill of some kind. You also have to have the tools to get the tree down in the first place. Even if buying milled lumber, you'll need tools to assist in making the conversion from raw wood to handle as painless as possible. Generally, having a band saw and at least some good quality rasps is kind of a minimum. You can do without and try chopping your waste wood out, but it'll add a lot of time and waste to the project. I often find myself using scraps from cutting handles for various small projects.
•Knowledge--Having the materials and tools means little if you don't know what makes a good handle. You could copy the shape of an off-the-shelf model, but then what are you bothering with all this extra work for? Knowing how to design an ideal handle for a given axe and intended use is necessary to really get the most out of the experience. So it's comparatively advanced compared to just popping an off-the-shelf stick in the eye and getting to work. If you're going to put the time in, you might as well end up with a handle that's shaped better than the one off the rack will.
•Time--You have to have the time available in order to do the work, and if working fresh from the tree that also means putting aside the billet for an extended period to dry out the wood properly. If you need that handle now then that may make that route unfeasible. Considerable forethought and planning is needed for that approach. But even working from lumber, you're going to be removing more material from a board to get a finished axe handle than you will be from a store-bought handle.
Add all of this up, and it's little wonder that back in the days before copy lathes most folks opted for straight handles--it was much more expedient than producing more complex forms. A lot of folks would buy handles from a local maker, but something fancy would cost ya' more. However, I firmly believe that in order to really maximize the performance of a given axe head, you pretty much need to have either designed your handle to match the head...or else designed your head to match typical store-bought handles. They're two halves of the same tool, and while the head may be doing the work it's the handle that's providing the interface between the user and the working end. It's kind of important to get that right.