Do the groups involved in geometrical isomerism have to be in similar planes to show geometrical isomerism?
To have geometrical isomers you need a structurally rigid bond. This bond can be double and this is a case where groups are in the same plane:



Or can be a cyclic compound where the  carbon-carbon single bond is restricted(2 examples below):

1st example:

 

2nd example:



Your textbook definition is incomplete and I wouldn't use the word "must" there.