NFC, Bluetooth, WiFi, etc isn't going to work. At least not with the Electrical Engineering skills of a mere mortal. I suggest that you think outside the box, so to speak.
A capacitive touch (captouch) screen might work, but would require significant changes to the existing software/firmware. Changes that might be beyond what a captouch screen manufacturer will be willing to tell you what/how to do it. Even so, here's how it works...
Normal Captouch is done with a microcontroller directly connected to the sensors. It scans the matrix, and builds up a 2-dimensional array of capacitiance values. It then processes that matrix to figure out things like the position of various fingers and converts that to gesture data which it sends via I2C or USB to the main CPU.
What you want is the raw matrix of capacitance values. To get that, you have to reprogram the microcontroller to send it to the main CPU instead of the gesture data. This is much easier said than done! The difficulty is going to be getting the information from the captouch manufacturer; ideally firmware source code and instructions on how to compile and download it. The second problem is that many (but not all) companies that make captouch microcontrollers will often not give support to people who are buying less than US$1mil/year from them.
There is also a risk that blocks of wood and other objects won't have enough of a capacitive response for the captouch sensors to detect. They are generally more sensitive than what we're used to, because their sensitivity is often turned down for normal finger type applications. But I cannot say if they are going to be sensitive enough for your requirements.
But let's say that you work past all of these issues... Then you have to identify the objects. In this case you will have to do it by shape and "captouch response". You could tell the difference between a wood and plastic cube because one would measure a higher capacitance. Or the difference between a cube and a moon from the shape itself.
Now, let's assume that the captouch thing doesn't pan out. The only other option that I can think of is by using a webcam. I am assuming that your screen is horizontal, like a table, so you can place things on it without them falling off. In this case, place a camera on the ceiling above the screen. Then, detect the objects in the camera image. Combine it with some software that displays calibration patterns on the screen.
Honestly, if I were designing this system, I would go straight to the camera version. I have developed captouch hardware and software and it is a pain in the #^@%&. The software itself isn't hard, but dealing with chip manufacturers is a pain, dealing with screen manufacturers is a pain. And dealing with buggy libraries and such is a pain. Not to mention that the odds of you getting assistance for either the chip/screen manufacture is almost zero.
