Wouldn't that mean that different versions of routing tables exists?
If yes, who controls what these routing tables contain?
Yes. Each router maintains its own routing table. Anycast is simply that the same network is advertised from multiple places. A router receiving multiple advertisements for the same network will choose which advertisement to place in its routing table based on the metrics of the routing protocol. This results in the closest (from the perspective of the routing protocol metrics) destination being used by a router.
For example, the primary way that BGP determines which of multiple destinations is the closest by the number of ASes in the AS_PATH attribute. If a router receives five advertisements for the
22.214.171.124/16 network, it will choose one to place in its routing table, and it will route the traffic destined for
126.96.36.199 to that network. If the site with that network goes down, the entry in its routing table will be withdrawn, and the next closest destination will be placed in its routing table.
Different routers in different locations will each make their own independent determinations as to which destination is closest. Anycast is simply advertising the same network from multiple sites and letting each router determine for itself where to send traffic destined for that network.