From my limited experience, it seems to have about the same role as the Western zodiac. Some people take it very seriously, just as there are those with a Western background who will say things like, "Of course the relationship didn't work out, they were both Leos!" A minority think it's pernicious nonsense and a plurality think it's harmless superstition or entertaining foolishness, or something with a grain of truth to it--maybe all three at once.

I can't remember it now, of course, but we learned in class how to say, "What year are you?" My Thai dictionary lists an equivalent expression. It's something people joke about, make reference to informally, even make insults about. I'd be surprised to find someone in this day and age who called off a wedding because the horoscope didn't work out, but not to find someone who earnestly ascribed marital problems to zodiacal incompatibility.

I was born in the Year of Dog and I've been told that since this is "my year", I should be able to accomplish great things. I wish I believe that were true, because then I could make it a self-fulfilling prophecy.