A lot of terminology gets thrown around about plywood quality. Terms used include "cabinet grade" and "furniture grade". Here's the scoop. You can open your own plywood store and call your stock anything you like.
Getting what you pay for can apply to plywood. There's been comments about how cheap can this or that be purchased. Then in the next breath, complaints about sanding through the veneer. I very rarely would buy any hardwood plywood from a big box store, but I don't make a habit of sanding the faces to the point of going through. I might cut all the panels needed and before assembly, just run a ROS with 150x or 180x just to clean it up. That would be to remove any handling oils and scrape markings from any machining.
Keep in mind that the big box stores carry a very high production of hardwood plywood, and it could come from anywhere. Their markings may not comply with what is considered standard for the industry. Plywood can be graded for appearance or purpose.
The Hardwood Plywood Veneer Association (HPVA) has established a grading that should be stamped on the edge. It is a two (2) character code. It refers mostly to the face and back veneer. For the faces, an alpha character (A-C) is used. "A" being the best, etc. The back veneer is a number (1-4), "1" being the best. Basically the grading follows the quality of the veneer, repairs and allowable defects. For example a sheet stamped "A1" would have a good face and a good back. A sheet stamped "A4" would have a good face and a back not intended to be visible, like a back to a cabinet. Grading any other way suggests that the plywood is of a construction or structural nature, not graded as "decorative".
As for the quality of the face veneers, the big box stores usually carry a "rotary cut" veneer faced plywood. That process utilizes the tree more productively. At a good hardwood supplier, there are several choices for the type of face veneer. To look more like the lumber you may want whats called "plain sliced" veneer. It is cut differently from the tree. Here is the cut explanation:
Knowing what to look for, with the type of veneer of a decent thickness might save you some headache and make you a better shopper.