Why does having identical groups on either side of the pi bond lead to no geometrical isomerism?
Hmm, you seem to be having a problem visualizing the structures in the case of 2 identical groups on one carbon. But never fear, 3D molecules are here.
Check out chloroethene:

It has two identical groups (hydrogens) on the left side. If you tried to make a geometrical isomer for this, you'd get:

But hey! That's just the previous image flipped upside down! That's not an isomer, it's the same thing!
Indeed it's the same molecule. These molecules can't have a geometrical isomer because it possess an plane of symmetry:

So there you go. Go ahead and visit this link if you want an interactive 3D model that you can spin around.

Added based on comments from @Zhe, there's a plane of symmetry in following angle,