"Is it possible to avoid using an RFID reader, and connect the microcontroller directly to the RFID?"
Using the RFID tag directly with a microcontroller without an RFID reader is useless, and often impossible.
Commercial RFID tags usually have only 1 connection to the outer world, and that's wireless. They draw their power supply from an RF carrier and send their data over that same carrier by loading it, so that the reader will see dips in the signal.
If you want to skip the RF, and directly connect, you may as well use a serial EEPROM which you read an ID from.
"the RFID tag needs to be on the same PCB as the display."
An RFID reader is the RF interface, which will pass the code received from the tag as an electrical signal to the microcontroller. So the microcontroller with its display is connected to the reader side, not the tag's.
For a reader this only makes sense. The display will change for each different tag the reader detects. If you connect the display to the tag it will always show the same code. That's unless you want to write to the tag to change its code.
The solution then would be to make your own tag. You'll need a microcontroller with a coil antenna connected that picks up the carrier, and which you will modulate. The other side of the microcontroller will drive the display. This will need much more power than can be drawn from the carrier, so you'll need at least a battery power supply. In which case an LCD is preferred over a LED display, because it uses less power.
Beth made her own RFID tag with just a small microcontroller and a coil. You'll need a bigger microcontroller with enough I/O to control the display.
An LCD module with integrated driver is easiest to use, though the one you chose is way overkill for just displaying numbers (unless you want to add fancy graphics too). It's a graphical display which AFAICS doesn't have a character generator, so you'll have to draw your digits on it yourself.
Anyway, whether you use a character display or a graphical one, the interface will be more or less the same: 8 (or 4) data lines and a few control lines. Any microcontroller with minimum 16 I/Os will do. Remember you'll also need at least one, maybe two I/O pins for talking to the RFID reader/writer.
Like most of these modules this one needs a 5V power supply. So choose a 5V microcontroller. If your favorite controller only works at 3.3V you can use level shifters to interface the two, but it's better to avoid them if not really needed.
You'll want a microcontroller with some EEPROM to store the code. Again, if yours doesn't have that you can use an external one, but you should be able to find a controller with the EEPROM on-chip.
In the Atmel ATMega series the ATMega8 is a controller that fits the bill.
See also this question.