So does that mean that all the kube-proxies inside a K8S cluster (running on each node) are mirror images?

Yea, They are instance of same image.

If so, why is a kube-proxy present on each node instead of a centralized one for entire cluster?

kube-proxy uses the operating system packet filtering layer if there is one and it’s available such as IPtable, IPVS. Otherwise, kube-proxy forwards the traffic itself. kube-proxy

Kube-Proxy is a k8s controller itself which watch the desired state(service & endpoints) of the cluster and make changes on the nodes, as it manage IPtabels (in use of iptable mode)

For this to be true, each kube-proxy will need to have complete information about all the services running in the cluster .....

there are following flags to set the behaviour of kube-proxy

--iptables-min-sync-period duration
The minimum interval of how often the iptables rules can be refreshed as endpoints and services change (e.g. '5s', '1m', '2h22m').
--iptables-sync-period duration     Default: 30s
The maximum interval of how often iptables rules are refreshed (e.g. '5s', '1m', '2h22m'). Must be greater than 0.

IMO, Decision of the connections (forwarding, accepting ) among pods on nodes should be made by node components rather than a central plane components. Besides, K8s Control plane (api-server, etcd) keep the desired state and current state of the cluster, so all of the controller can reconcile according to their set behaviour.