There was a glitch in the system previously where you could buy more than one of a birthstone, but it was unintentional, and they fixed it but anybody who had exploited the glitch had the extra birthstones removed with no refunds.
So no, you aren't able / supposed to own more of one birthstone and I don't think the year matters, it's still the same collectible.
im pretty sure if someone gifts you a birthstone you already had it would have some sort of error message appear. but the birthstone will not disappear.