I don't know if it's normal. My locally-built cabinets do have an entire back but I don't think it's needed. That looks like a pretty sturdy cabinet box.

Perhaps request the cabinet guy to tack a back on? Just because it isn't necessary doesn't mean you can't have it. :)

There is nothing in that picture that would be acceptable. That is a backyard garage woodshop guy. Avoid those types at all costs.

All my cabinets have full boxes regardless if they are cabinets or drawers. Over the years I've hired 2 different custom cabinet builders. One was recommended to me by a local designer. After he retired, the second one was recommended to me by my favorite painter. So try to find a local designer, a really good painter, or ask your neighbors for referrals.

In both cases, I visited their shops. Neither one was in a personal garage or backyard! They were workshops with lots of equipment and projects in process, and they each showed me their building process, finished cabinetry, and furniture.

I'd recommend that you visit the shops of whoever you are considering and have them show you everything they do from wood selection, to construction of the boxes, to door/drawer fronts, to what type/brand hinges and drawer glides, and anything else you can think of. Some cabinet builders sub-out the door/drawer fronts. If that's the case for the builders you talk to, then you'll want to see some examples of the fronts.

We put a full backs in all our drawer cabinets, but I have seen many that don't to save a couple of dollars. If that is something you want, tell them. They are a custom cabinetmaker and it would be no problem.