We have custom cabinetry in our log home. They draw a lot of comments because of what their made of: hickory with a clear stain. We had to special order them because they were fairly expensive and no store carried them in stock. Home Depot does stock some hickory cabinets but there's not a large selection, they're much more cheaply made (the boxes are particle board while mine are solid wood), and the hinges and drawer slides are cheap Chinese stuff.
We too have hickory with natural finish in our garage!
Just a few minor corrections. You'd be hard pressed to find cabinet carcasses made of solid wood. And if you found someone to build them for you, you can't get it from either of 2 major cabinet manufacturers (or their subsidiaries). You'd have to go to what they call a "bubba shop," or a small mom and pop operation locally. The price will be astronomical.
The cabinets in my garage are from omega, one of the finer cabinet companies, under the masterbrand companies. Carcasses are made mostly of ply, and they use some "particle board" as well.
Home Depot most likely orders very few custom cabinets, most have to go through actual designers. But, the big box stores can "special order" standard made cabinets.
I personally don't have a problem with particle board as the carcass in most situations. It saves money and if used within the intended scope (e.g. not stashing all your gold bars in a stand alone wall cabinet), they'll last forever if not abused.
The OP's cabinets are most definitely oak. Personally the finish job is off putting to me, but that can be changed. Just don't forget to fill the pores if you decide to paint.