I vividly remember the moment a friend pointed out that perhaps the reason my husband and I were butting heads =early in our marriage, was because we were both oldest siblings. I could have fallen on the floor, that aha moment hit me so hard. I had never given so much as a second thought to birth order, but once my friend mentioned it, I realized there are actually reasons you should care about birth order. It isn't the end all be all of procreation, by any means, but it seems as though it can dictate personality traits for you, as a parent, as well for your kids.
Growing up, I remember people casually describing my sister as the "classic middle child." She marches to her own drum, and her tantrums about what clothes she wanted to wear were epic. I remember my mom devising systems to avoid or correct her behavior, including complicated sticker charts and reward systems. My sister and I were also absolutely aware that our youngest brother was the "baby," and nicknamed him "Angel Eyes." The kid could never do any wrong. To this day, I don't think he's ever been in trouble.
I don't remember ever hearing any conversations about my firstborn child status, but perhaps people just didn't want to say anything to my face. Of course, birth order doesn't explain every single personality trait, in either you or your kids. However, I think it can help you understand certain reactions you or your kids may have. At the very least, it's interesting to think about as you're raising multiple kids.