Try it with coins: First, flip a nickel and a dime. There quite clearly are four possibilities: (N=H, D=H), (N=H, D=T), (N=T, D=H), and (N=T, D=T). Each, also quite clearly, has a probability of 1/4; so the probability of two heads is 1/4, the probability of two tails is 1/4, and the probability of one of each is 1/2.
Then repeat the same experiment with two absolutely identical quarters. There now seem to be three combinations we can discern: (Both H), (Both T), and (one H and one T). But do you really think the odds change to 1/3 for each combination just because the coins are now identical? I certainly hope not. Even though you can't see the difference between the coins, they still are different coins, and the ways that the different combinations can occur still depend on that difference.
When we count possibilities for two-child families, age-order matters only because we have to acknowledge that the two children are different from each other. Even if we aren't given any information that distinguishes them, that difference still matters.
But as for the probability paradox you mentioned, the answer depends on how you "know" that one is a boy. If it is because you asked if the family had any boys, then every BB, BG, or GB family would answer "yes" and the answer is 1/3. But if you learned that fact incidentally, then you could have learned a BG of GB family has a girl. You can only count half of them, and the answer is 1/2.