Mode 7 essentially is a primitive 3D accelerator, it renders a single textured plane with various transforms. I think in principle it's just as "real 3D" as a modern GPU, it's just very very limited and less accurate.
The cool 3D hack comes from co-processors in the cartridge: Games like Super Mario Kart use extra chips than can quickly compute new mode 7 parameters per scanline: If you change the zoom and translation of mode 7 on every line (but keep the rotation the same), you can get the typical tilted 3D look by having the upper lines zoomed out (smaller) and the lower lines zoomed in more (bigger).
This "change the zoom on each scanline" trick is very different from what modern GPUs do and not an arbitrary transform, e.g. with the scanline trick you can only rotate the plane around the horizontal axis, but not around the vertical one.
F-Zero managed to do the “change zoom on each scanline” trick without a coprocessor in the cartridge, didn’t it?
Mode-7 scaling can be modified per scanline to get the textured-plane effect seen in many games (e.g. F-Zero), without any additional hardware.
Super Mario Kart needed the co-processor for determining things like how to draw the karts depending on their position and distance from the player. Pilotwings used the same co-processor for similar things (essentially, tracking objects and perspective in 3D space).