The basis for all of this is that Gems have [i]elemental magic[/i]. They are born with an elemental affinity and it is intrinsically part of them, whether they like it or not. Therefore, it follows that the element would affect them in some way. 

What I am [i]not[/i] including is actual, real-life shit. Science is a good starting point, but at the end of the day, this is fantasy. I'm not writing an anthropological dissertation on the societal interpretations of ethnic traits. 

The basic idea is this: It's more common for your personality to associate with your element than not. This does not rule out exceptions, but it also doesn't mean those exceptions are considered normal. 

Now, there are two factors that cause this: genetic affinity and societal influence. (No, I am not going to debate nature vs. nurture with you. That's still hotly debated by people with much more education than I)

Genetic affinity is the fact that Gems are born with an element. Unlike the Drakken, whose elements don't really affect them (for lore reasons I'm not getting into in this explanation), Gems are much more tuned to their elements by nature (hence their increase in control). This means that, more often than not, some of the vague traits of that element will seep into the Gem. For example, Fire gems tend to be more reactive and Water gems tend to be more calm. These do not so much affect specific personality traits as they do general disposition. 

Then there's societal influence. The Gems (due to that aforementioned disposition link) associate certain values and traits with certain elements, and this is where it becomes more of a cultural thing than directly elemental. Water Gems as a group hold such values as wisdom, temperance, and persistence (and to an extent, forethought and memory) in high regard. Likewise Earth gems tend to value solidarity, loyalty, empathy, and compassion. Fire gems could appreciate resolve, action, confidence, etc., and Air gems things like flexibility, innovation, freedom of expression, adventurous-ness etc. Definitely not an exhaustive list but you get the idea. In any case, when you're raised with certain sets of morals and ideals (and role models who embody them), you come to value and appreciate them, which in turn affects how you think and act. (This is where you can get some cool cultural variations between Gems raised in mono-elemental villages and mixed-elemental cities)

Of course, there are always outliers. It's entirely possible to get an airheaded water gem with a quick temper, but they would probably get looked at funny by others of their community and feel as if they don't really fit in. They may feel disconnected from their element. The level of affinity one feels for their element may vary too - perhaps a fire gem feels connected to fire and feels the disposition pull, but is still shy.

Now, this doesn't mean that the elemental affinities aren't common. It would be uncommon to find a gem like the water one above, but of course it could happen. This is why the elemental affinity thing isn't really a stereotype; it has more truth and substance to it than a stereotype. It's more of a very general average - it would be more common to find a gem that mostly fits their element than one who clearly does not. Kind of like how Caucasian people tend to be medium-height. It's not a stereotype because it's true for a large portion of the population, but you still get some very short and very tall Caucasians.