First of all, there are so many members of this forum who are more qualified and able to talk about this than me, and for what it's worth, I have at times disagreed with many of them in the past, and in some of those cases no longer disagree with them. So take what I say at face value, I guess?
First, try not to think of things in terms of defining whether Person X is sexist. This makes things difficult for a few reasons, not the least of which is that if Person X is part of the conversation he or she will probably become agitated and defensive and other things that don't make it easier to have a productive conversation. Try to instead think in terms of actions and attitudes that have sexist effects or outcomes. Let's discuss things from the fairly pragmatic perspective of, "What improves the situation of the disadvantaged group?" rather than asking, "Which people are bad, sexist people?"
So, let's talk about the connection between using a term like "chick drink" and sexism, rather than the connection between using the term and being A SEXIST ASS WHO SUCKS or any other such charged, antagonistic perspective. The term does two things; on one level, it defines a certain group of drinks by common characteristics which have to do with the drinks. It's kind of ambiguous, and we've spent a good amount of this thread trying to pin that definition down. On another level, it defines that same group of drinks by one characteristic which has nothing to do with drinks, but with stereotypical masculinity and femininity. Not medical or genetic or chemical maleness and femaleness, mind you.
Now, if there's a medical or genetic or chemical way to classify drinks as being proper for one gender or another, I'm not aware of it. More importantly, neither are the people who use the term "chick drink". What we're doing is deciding on some way to associate drink characteristics with those gender stereotypes. Can we agree that the perpetuation of gender stereotypes is harmful, at the very least, to people who fall outside of them? Because if we can, then it shouldn't be too much of a leap to arrive at the conclusion that classifying drinks, or anything really, according to those gender stereotypes is similarly harmful. Is it a big deal? Not to the extent that gender discrimination in hiring practices or pay scale is a big deal, perhaps. But it's still an issue because it still has an impact.
tl;dr: When you call something a "chick drink" you divide things by gender in a way that is not based on actual gender, but on stereotypes, thereby perpetuating and reinforcing those stereotypes through your language. Furthermore, the common perception of the term "chick drink" is that it refers particularly to drinks that are weak, impotent, barely drinks at all. They have less alcohol, or they are designed to seem as though they have less alcohol. They are sweet, they are easy to drink, they are for people who are unable to take other drinks. So the association you're reinforcing is one that pairs women with weakness, impotence, inability, and a sense of not counting; a chick drink has so little alcohol, it might as well not be a drink at all. A woman has so little of what makes a person worth something, she might as well not be a person at all. This is an extreme position, sure, but it's present in these little turns of phrase, only in tiny amounts. This doesn't necessarily go through someone's conscious mind when they talk about a "chick drink", but it does affect their perception.
And now I'm going to press "preview" and see whether I've been ninja'd yet by one of the folks with more practice in this sort of discussion, and who probably takes far less time to say the same thing better... (Oh hey, it's my wife. Hi wife! You're one swell chick.)