I came across the following lines in which the word "genial"
and "geniality" are used. Should it be "geniuses" and "genius"
respectively, instead? "genial"/"geniality" has a totally
different meaning and does not convey the sense of "genius".
If a child succeeds in overcoming the established norms, we say that “this is a genius”. But in fact we do not realize that “we are all genial”. If applied correctly, suggestopedia establishes a communication at the level of our individual “geniality”
This is a false friend, much loved by Fench and Spanish students. But if you look in a big enough dictionary, you'll see that it was used - fairly recently (1850) in the sense 'having the characteristics of genius'.
ps - 1850 is the date given by the Shorter Oxford English Dictionary for a first sighting, but this usage is one that I don't remember hearing.