This isn't really an answer, but I thought it was interesting and it's too long to fit in a comment:
(This is more of an answer now; see the edit at the bottom.)
Oddthinking's answer inspired me to do some digging of my own, and I too was met with a lack of immediately relevant academic research. I therefore decided to go about this indirectly by looking at the fruit and vegetable intake of children. This 2009 study by Lorson, Melgar-Quinonez, and Taylor found that 74.1% of children in the US between the ages of 6 and 11 do not meet the recommended daily intake of fruit (on average, these children consume 71.5% of the recommended daily intake). Likewise, among the same group of children, 83.8% did not meet the daily recommended intake of vegetables (on average, consuming 58.3% of the recommended intake). While this study only measured the quantity of intake, I think it is at least plausible to hypothesize that a lack of quantity should be correlated to lack of variety (and thereby lack of familiarity).
This 2009 study by Coyle, Potter, Schneider, May, Robin, Seymour, and Debrot surveyed a number of students of different ages in Mississippi, USA, as to their familiarity with different types of fruits and vegetables. The survey asked the students if they were familiar with or had ever eaten 13 common fruits and 7 common vegetables. The 5th grade students, on average, were familiar with only 85% of the fruits and only 61% of the vegetables. The familiarity did increase slightly with age, with a value of 91% for fruits and 68% for vegetables by grade 10. Therefore, I think it is reasonable to hypothesize that 6 year olds, although not studied, would have had no better (and most likely worse) familiarity than the 5th graders.
I need to clarify my summary of the survey method from the second study, as the paper is a little ambiguous. After having re-read the paper, it seems as if the survey listed 13 fruits and 7 vegetables and then asked something like, "List all of the fruits/vegetables you can think of that you have ever eaten, including but not limited to those above." The percentages I quoted are just the number of fruits or vegetables the students listed divided by the number of examples the survey listed. Therefore, if a student received a "score" of 0.77 for fruits, that means that he or she could only recall the names of 0.77*13 = 10 different fruits that he or she had eaten in his or her lifetime. It is therefore possible to have "scores" higher than 1.0. Off the top of my head, I can easily think of at least two dozen different types of fruit that I've eaten throughout my life, which would have given me a score of 24/13 = 1.85.