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Hmm ... I had always understood the reference to "a colorful past" (in the anime I have seen it translated as "you also have a lot going on") to refer to the difficulty she has with her father not supporting her mahjong play.
The Achiga girls are a somewhat related instance of this - that is, she was separated from a group of friends who shared the same love for the game as she did because of her parents' (specifically mother I believe) job transfers.
In many ways I think it is a sort of emphasis on mahjong, yes - but also on her friendship. Her father especially seems to be (and understandably from a certain perspective) concerned primarily for Nodoka's future and wish to provide for it through education. The mother seems to be rarely around.
But Nodoka, it seems to me, is ultimately a shy, quiet, bookish, exceedingly intelligent and intense girl who has interests that are not commonly shared by others her age (i.e. mahjong). Mahjong is important to her, of course, but so are people and friends who she feels she can be open and herself with.
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