All we know in Star Wars is that Leia inserted a disc into R2-D2. At no point, do they explicitly say he has access to the information on the disk, or that it was uploaded into him. He never displays it as a hologram. When we see the blueprints used later, it's via the Rebel headquarters computers/display system.
This is almost exactly like The Force Awakens, where Poe places a
usb thumb drive data stick in BB-8's storage compartment.
He just drops it in. He doesn't insert/connect it. Of course, somehow, BB-8 can access the data, and does show it on hologram, meaning he can use the map. BB-8 being almost 60 years more advanced than R2-D2 though, considering they have upgraded from floppy disks to usb sticks. Maybe they finally invented wireless usb or Bluetooth.
When R2-D2 has to access the Death Star computer to find where the Tractor Beam power source is located, he actually talks to the computer, and it tells him where it is. R2 doesn't access the data directly.
Remember, pretty much all computers in Star Wars are sentient A.I. to some degree. As C3PO says:
C-3PO: Noisy brute. Why don't we just go to lightspeed?
C-3PO: We can't? How would you know the hyperdrive is deactivated?
C-3PO: The City's central computer told you? R2-D2, you know better than to trust a strange computer.
It's a conversation between two distinct persons (R2-D2 and the Ship), not two computers accessing files.
Basically, R2-D2 has the map, in the sense of being a really sassy, expensive CD Case.