Nitpick (I blame Amazon for their horrible naming): Aurora and RDS are separate products.
What you’re asking for can mostly be pieced together, but no, it doesn’t exist as-is.
Failover: this has been a thing for a long time. Set up a synchronous standby, then add a monitoring job that checks heartbeats and promotes the standby when needed. Optionally use something like heartbeat to have a floating IP that gets swapped on failover, or handle routing with pgbouncer / pgcat etc. instead. Alternatively, use pg_auto_failover, which does all of this for you.
Clustering: you mean read replicas?
Volume-based snaps: assuming you mean CoW snapshots, that’s a filesystem implementation detail. Use ZFS (or btrfs, but I wouldn’t, personally). Or Ceph if you need a distributed storage solution, but I would definitely not try to run Ceph in prod unless you really, really know what you’re doing. Lightbits is another solution, but it isn’t free (as in beer).
Cross-region replication: this is just replication? It doesn’t matter where the other node[s] are, as long as they’re reachable, and you’ve accepted the tradeoffs of latency (synchronous standbys) or potential data loss (async standbys).
Metrics: Percona Monitoring & Management if you want a dedicated DB-first, all-in-one monitoring solution, otherwise set up your own scrapers and dashboards in whatever you’d like.
What you will not get from this is Aurora’s shared cluster volume. I personally think that’s a good thing, because I think separating compute from storage is a terrible tradeoff for performance, but YMMV. What that means is you need to manage disk utilization and capacity, as well as properly designing your failure domain. For example, if you have a synchronous standby, you may decide that you don’t care if a disk dies, so no messing with any kind of RAID (though you’d then miss out on ZFS’ auto-repair from bad checksums). As long as this aligns with your failure domain model, it’s fine - you might have separate physical disks, but co-locate the Postgres instances in a single physical server (…don’t), or you might require separate servers, or separate racks, or separate data centers, etc.
tl;dr you can fairly closely replicate the experience of Aurora, but you’ll need to know what you’re doing. And frankly, if you don’t, even if someone built a OSS product that does all of this, you shouldn’t be running it in prod - how will you fix issues when they crop up?