First off, here's a picture (source: NASA) of the inside of the MIR node with something connected in every direction:
There's also a video here where the node is traversed, but I can't find any pictures from before all the ports were occupied.
It's not particularly spacious, but in the picture only one cone-shaped drogue hatch cover is clearly visible, as well as one cover near the camera (can't tell its geometry) and what looks like a flat cover through the far hatch. The specs of the base module (from Mir Heritage Hardware, page 104) that specifies 2 docking and 4 berthing ports (docking were on the ends of the module, berthing were on the sides of the node, difference being that berthing ports didn't have an automatic docking system). All ports appear to use the SSVP docking system.
Procedure for connecting a module to a berthing port (also from Mir Heritage Hardware) involved docking the module to the axial port of the node, then using a manipulator arm to move it to a berthing port. I don't see any indication that a different docking mechanism was used, meaning that each berthed node would have needed a drogue port to connect to. I can't currently find a source, but I believe that once connected the probe/drogue could be removed to free up space around the transfer tunnel (and the pictures/video are consistent with this).
That said, comparing the size of the drogue port cover pictured and the size of the node the drogue cover would take up interior volume, but it would also be on a face of the node that isn't in use. I don't think it would intrude far enough into the node to be much of an obstruction.
To directly answer the question: No, it doesn't appear that the docking ports consume much interior space. I think the real reason they didn't include more docking/berthing ports in Zarya/Zvezda is a combination of the close proximity of solar panels (zarya/zvezda pictures) on the sides and a desire not to add complexity or holes in the space station that weren't expected to be used.