Well, if you google 'birthstone list' and look at the images, you'll see that some months don't have just one set stone. For example, I see 'citrine or topaz' for November.
However, there's still a problem. Both Amethyst and Onyx are gemstones for February and no other month.
So, based on googling them one at a time with the keyword "birthstone", here is what came up:
Amethyst - February
Pearl - June
Garnet - January
Diamond - April
Emerald - May
Ruby - July
Sapphire - September
Moonstone - June
Topaz - Orange variant is November, blue variant is December
Citrine - November
Onyx - February
Zircon - December
As you can see, there are plenty of repeated months and some months are missed (October for example is represented by Tourmaline or Opal, which both do not make an appearance here).