I wonder whether that's a mistake (under "koala bear"):
(Do I know my marsupials or do I know they are marsupials?)
They probably wrote it the way they meant it, but it did make me do a double take for a second.
It was written as intended. It's strange to think of it in that possessive way, but it's common usage for indicating people are very much in the know about a given subject (ex: Those sailors really know their tides.). I'm not sure of why this is, but if I had to guess I would say it's meant to suggest the indicated party's knowledge is such that whatever falls into the scope of their understanding belongs to said party on some foundational level in the way that you know the back of your hand so well because it is a part of you and, as a result, is yours. That's just educated guessing, though - no research to back it up or anything. ;p