Anycast route election is done by the connecting router (typically the clients ISP), and it works over bgp like normal route advertising - it's transparent to your applications and they only get directed to one endpoint destination (typically the one with the lowest latency, congestion, or cost). Once the router gets a request from a client for a destination, it does the same thing it usually does and checks the possible routes and chooses the "best" based on the rules defined by the owner of the router. The difference here is that not all of those proposed routes make it to the same physical destination. There is some stickyness to make sure you're arriving at the same destination for the duration of that connection.
Really it's like asking your favorite maps app where McDonalds is - it'll know to give you results close to you, since it knows there are many McDonalds. It would make sense for you to choose the closest, but it's not enforced by McDonalds. Similarly, if I do the same and we live in different cities, we'll get different results. The end result is that we both got to different physical places by specifying the same thing, and it's up to McDonalds to make sure the menu is the same :)